13126:
20:
12712:
5768:
856:
12182:
263:
5430:
6133:
12707:{\displaystyle {\begin{aligned}r&\geq q({\mathcal {Q}})-q({\mathcal {P}})\\&=\sum _{i,j}{\frac {|V_{i}||V_{j}|}{n^{2}}}\mathbb {E} |d(W_{i},W_{j})-d(V_{i},V_{j})|^{2}\\&\geq \sum _{i,j}{\frac {1}{4|P|^{2}}}\mathbb {E} |d(W_{i},W_{j})-d(V_{i},V_{j})|^{2}\\&={\frac {1}{4|P|^{2}}}\mathbb {E} \sum _{i,j}|d(W_{i},W_{j})-d(V_{i},V_{j})|^{2}\end{aligned}}}
3206:
5763:{\displaystyle q({\mathcal {Q}})=\sum _{(i,j)\in ^{2}}q({\mathcal {Q}}_{V_{i}},{\mathcal {Q}}_{V_{j}})=\sum _{(V_{i},V_{j}){\text{ }}\varepsilon {\text{-regular}}}q({\mathcal {Q}}_{V_{i}},{\mathcal {Q}}_{V_{j}})+\sum _{(V_{i},V_{j}){\text{ not }}\varepsilon {\text{-regular}}}q({\mathcal {Q}}_{V_{i}},{\mathcal {Q}}_{V_{j}})}
2057:
4775:
4233:
6540:
1770:
11084:
The corollary explores deeper the small energy increment. It gives us a partition together with subsets with large sizes from each part, which are pairwise regular. In addition, the density between the corresponding subset pairs differs "not much" from the density between the corresponding parts.
5868:
10435:. By equitizing in each round of iteration, the proof of regularity lemma could be accustomed to prove the equitable version of regularity lemma. And by replacing the regularity lemma with its equitable version, the proof above could prove the equitable version of strong regularity lemma where
2911:
3957:
13202:
extends the Szemerédi regularity lemma, by revisiting it from the perspective of probability theory and information theory instead of graph theory. Terence Tao has also provided a proof of the lemma based on spectral theory, using the adjacency matrices of graphs.
724:
13190:
gave a different version and extended it to hypergraphs. They later produced a different construction due to Alan Frieze and Ravi Kannan that uses singular values of matrices. One can find more efficient non-deterministic algorithms, as formally detailed in
7732:
886:
to show that this process stops after a bounded number of steps. To do this, we define a measure which must increase by a certain amount in each step, but it's bounded above and thus cannot increase indefinitely. This measure is called 'energy' as it's an
2922:
1782:
10173:
7323:
11960:
1381:
7387:
A different notion of regularity was introduced by Frieze and Kannan, known as the weak regularity lemma. This lemma defines a weaker notion of regularity than that of Szemerédi which uses better bounds and can be used in efficient algorithms.
4522:
3993:
13174:
that the regular pairs in Szemerédi regularity lemma behave like complete bipartite graphs under the correct conditions. The lemma allowed for deeper exploration into the nature of embeddings of large sparse graphs into dense graphs.
6337:
12883:
6128:{\displaystyle \sum _{(V_{i},V_{j}){\text{ }}\varepsilon {\text{-regular}}}q(V_{i},V_{j})+\sum _{(V_{i},V_{j}){\text{ not }}\varepsilon {\text{-regular}}}q(\{A^{i,j},V_{i}\backslash A^{i,j}\},\{A^{j,i},V_{j}\backslash A^{j,i}\})}
1525:
9618:
2650:
5089:
8667:
3349:
2247:
10985:
9454:
13058:
9382:
8056:
3783:
3664:
226:
10552:
8295:
4448:
424:
67:
To state Szemerédi's regularity lemma formally, we must formalize what the edge distribution between parts behaving 'almost randomly' really means. By 'almost random', we're referring to a notion called
2393:
2324:
1240:
1167:
11460:
9678:
1095:
4349:
10351:
7489:
7352:, by considering edge edits instead of only edge deletions. This was proved by Alon, Fischer, Krivelevich, and Szegedy in 2000. However, this required a stronger variation of the regularity lemma.
4867:
3508:
1446:
557:
7170:
12187:
10255:
6329:
11036:
3271:
10642:
8885:
6942:
1517:
588:
9718:
8925:
9968:
9865:
3775:
11178:
10839:
8848:
8600:
7379:-regularity is trivially satisfied. Even though the result seems purely theoretical, some attempts have been made to use the regularity method as compression technique for large graphs.
12090:
3201:{\displaystyle \mathbb {E} =\sum _{i=1}^{k}\sum _{j=1}^{l}{\frac {|U_{i}|}{|U|}}{\frac {|W_{j}|}{|W|}}d(U_{i},W_{j})^{2}={\frac {n^{2}}{|U||W|}}q({\mathcal {P}}_{U},{\mathcal {P}}_{W})}
11561:
7552:
11784:
9533:
11500:
11265:
9914:
8708:
6258:
5809:
2052:{\displaystyle q({\mathcal {P}})=\sum _{i=1}^{k}\sum _{j=1}^{k}{\frac {|V_{i}||V_{j}|}{n^{2}}}d(V_{i},V_{j})^{2}\leq \sum _{i=1}^{k}\sum _{j=1}^{k}{\frac {|V_{i}||V_{j}|}{n^{2}}}=1}
9761:
9094:
12961:
9227:
5287:
5241:
3583:
13115:
11074:
10877:
10751:
9271:
11216:
10682:
9302:
9008:
8751:
8116:
7792:
2129:
2098:
13303:
A stronger variation of the regularity lemma was proven by Alon, Fischer, Krivelevich, and
Szegedy while proving the induced graph removal lemma. This works with a sequence of
10018:
6841:
6714:
7904:
6684:
6570:
2622:
10023:
7178:
7055:
11984:
11813:
11808:
11693:
11402:
11333:
11289:
10481:
10457:
10406:
8973:
8949:
8807:
8559:
8508:
8482:
8458:
8434:
8410:
8383:
8359:
8170:
7975:
7928:
7843:
7544:
6594:
6220:
5860:
5335:
5311:
4964:
4940:
4770:{\displaystyle Var(Z)=\mathbb {E} )^{2}]\geq {\frac {|U_{1}|}{|U|}}{\frac {|W_{1}|}{|W|}}(d(U_{1},W_{1})-d(U,W))^{2}>\varepsilon \cdot \varepsilon \cdot \varepsilon ^{2}}
3607:
3446:
3422:
1474:
11137:
10798:
9190:
9035:
8778:
8535:
6988:
5175:
3554:
1252:
13321:
13278:
13231:
10578:
7377:
7008:
6734:
6657:
6614:
5195:
4916:
3723:
578:
9149:
4514:
4481:
11378:
9784:
8193:
8079:
7997:
7755:
7512:
13323:
instead of just one, and shows that there exists a partition with an extremely regular refinement, where the refinement doesn't have too large of an energy increment.
12914:
12175:
6634:
4794:
3395:
7427:
6196:
5368:
4228:{\displaystyle Var(Z)=\mathbb {E} -\mathbb {E} ^{2}={\frac {n^{2}}{|U||W|}}\left(q\left(\{U_{1},U\backslash U_{1}\},\{W_{1},W\backslash W_{1}\}\right)-q(U,W)\right)}
13258:
12144:
12117:
12038:
12011:
11743:
11720:
11669:
11642:
11615:
11588:
9811:
6815:
6163:
5836:
5422:
5395:
5129:
5018:
4991:
3703:
2567:
2520:
971:
912:
9123:
6535:{\displaystyle \sum _{(i,j)\in ^{2}}q(V_{i},V_{j})+\sum _{(V_{i},V_{j}){\text{ not }}\varepsilon {\text{-regular}}}\varepsilon ^{4}{\frac {|V_{i}||V_{j}|}{n^{2}}}}
4896:
10281:
3375:
13298:
11309:
10702:
10433:
9055:
8335:
8315:
7948:
7819:
7095:
7075:
6881:
6861:
6774:
3985:
2642:
2540:
2493:
2473:
2453:
2433:
2413:
2169:
2149:
495:
475:
9486:
7875:
1765:{\displaystyle q({\mathcal {P}})=\sum _{i=1}^{k}\sum _{j=1}^{k}q(V_{i},V_{j})=\sum _{i=1}^{k}\sum _{j=1}^{k}{\frac {|V_{i}||V_{j}|}{n^{2}}}d(V_{i},V_{j})^{2}}
12719:
6749:
If we have enough information about the regularity of a graph, we can count the number of copies of a specific subgraph within the graph up to small error.
13658:
Fiorucci, Marco; Pelosin, Francesco; Pelillo, Marcello (February 2020), "Separating structure from noise in large graphs using the regularity lemma",
9763:
the regular and refinement conditions hold. The energy condition holds trivially. Now we argue for the number of parts. We use induction to show that
2906:{\displaystyle \mathbb {E} =\sum _{i=1}^{k}\sum _{j=1}^{l}{\frac {|U_{i}|}{|U|}}{\frac {|W_{j}|}{|W|}}d(U_{i},W_{j})={\frac {e(U,W)}{|U||W|}}=d(U,W)}
9538:
8142:, however an algorithmic version of the weak regularity lemma gives an efficient algorithm for approximating the max-cut for dense graphs within an
274:-regular if, whenever you take a large subset of each part, their edge density isn't too far off the edge density of the pair of parts. Formally,
5026:
55:
in 1975 and for general graphs in 1978. Variants of the lemma use different notions of regularity and apply to other mathematical objects like
7334:
8607:
39:
can be partitioned into a bounded number of parts so that the edges between parts are regular. The lemma shows that certain properties of
13206:
It is not possible to prove a variant of the regularity lemma in which all pairs of partition sets are regular. Some graphs, such as the
3276:
2174:
10883:
9387:
3952:{\displaystyle q\left(\{U_{1},U\backslash U_{1}\},\{W_{1},W\backslash W_{1}\}\right)>q(U,W)+\varepsilon ^{4}{\frac {|U||W|}{n^{2}}}}
11218:. The full version can be proved by picking more subsets from each part that are mostly pairwise regular and combine them together.
14371:
12966:
9307:
8007:
13233:-regular partition to require that the vertex sets all have the same size, while collecting the leftover vertices in an "error"-set
4800:
Now we can prove the energy increment argument, which shows that energy increases substantially in each iteration of the algorithm.
817:
for the number of parts in the partition of the graph given by the proofs of
Szemeredi's regularity lemma is very large, given by a
14173:
8172:
additive error. These ideas have been further developed into efficient sampling algorithms for estimating max-cut in dense graphs.
133:
14706:
10499:
8242:
4354:
354:
13802:
2329:
2260:
1176:
1103:
11407:
9623:
995:
14500:
4241:
3612:
14838:
13883:
10286:
7432:
4810:
3451:
1389:
500:
7100:
827:. At one time it was hoped that the true bound was much smaller, which would have had several useful applications. However
13635:
10178:
6263:
719:{\displaystyle \sum _{(V_{i},V_{j}){\text{ not }}\varepsilon {\text{-regular}}}|V_{i}||V_{j}|\leq \varepsilon |V(G)|^{2}}
10990:
3214:
10583:
8853:
6886:
1479:
14822:
14806:
14782:
14726:
14445:
14337:
14282:
13434:
13375:
13159:
9683:
8890:
14392:
Alon, N.; Duke, R. A.; Lefmann, H.; Rödl, V.; Yuster, R. (1994), "The algorithmic aspects of the regularity lemma",
13125:
9919:
9816:
3728:
11142:
10803:
8812:
8564:
7727:{\displaystyle \left|e(S,T)-\sum _{i,j=1}^{k}d(V_{i},V_{j})|S\cap V_{i}||T\cap V_{j}|\right|\leq \epsilon |V|^{2}}
14881:
14468:
Frieze, Alan; Kannan, Ravi (March 1999), "A simple algorithm for constructing Szemerédi's regularity partition",
14430:
37th Annual
Symposium on Foundations of Computer Science, FOCS '96, Burlington, Vermont, USA, 14–16 October, 1996
13518:
12043:
8126:
One of the initial motivations for the development of the weak regularity lemma was the search for an efficient
13918:
11508:
11751:
9491:
742:
11465:
11224:
9870:
8673:
6225:
5776:
14592:
Austin, Tim (2008), "On exchangeable random variables and the statistics of large graphs and hypergraphs",
14428:
Frieze, Alan M.; Kannan, Ravi (1996), "The regularity lemma and approximation schemes for dense problems",
9726:
9060:
8196:
442:, require many pairs of partitions (but a small fraction of all pairs) to be irregular. So we shall define
36:
12919:
9195:
8227:
in 2000. Intuitively, it provides information between non-regular pairs and could be applied to prove the
5246:
5200:
14886:
13331:
13063:
11041:
10844:
10710:
9232:
11183:
10647:
10168:{\displaystyle |P_{i+1}|\leq M(\epsilon _{|P_{i}|})|{\mathcal {P}}_{i}|\leq M(\epsilon _{|M_{i}|})M_{i}}
9276:
8982:
8725:
8084:
7760:
7318:{\displaystyle \left(\prod _{\{i,j\}\in E(H)}d(V_{i},V_{j})\right)\left(\prod _{i=1}^{k}|V_{i}|\right).}
6639:
Now, starting from any partition, we can keep applying Lemma 3 as long as the resulting partition isn't
2103:
2072:
11955:{\displaystyle {\frac {r}{|P|^{4}}}{\binom {n}{2}}\leq \epsilon _{0}{\frac {|V_{i}||V_{j}|}{3|P|^{2}}}}
9973:
6820:
6689:
7880:
6662:
6548:
2572:
1376:{\displaystyle q({\mathcal {P}}_{U},{\mathcal {P}}_{W}):=\sum _{i=1}^{k}\sum _{j=1}^{l}q(U_{i},W_{j})}
7013:
3559:
14129:
13137:
13136:
first introduced a weaker version of this lemma, restricted to bipartite graphs, in order to prove
11965:
11789:
11674:
11383:
11314:
11270:
10462:
10438:
10356:
8954:
8930:
8788:
8540:
8489:
8463:
8439:
8415:
8391:
8364:
8340:
8145:
7956:
7909:
7824:
7517:
7342:
6575:
6201:
5841:
5316:
5292:
4945:
4921:
3588:
3427:
3403:
1455:
52:
14406:
11096:
10757:
9157:
9013:
8756:
8513:
8204:
7338:
6947:
5134:
3513:
14341:
14286:
13306:
13263:
13216:
13163:
10557:
7362:
6993:
6719:
6642:
6599:
5180:
4901:
3708:
563:
14867:
Szemerédi's regularity lemma (Formal proof development in
Isabelle/HOL, Archive of Formal Proofs)
13990:
Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976)
9128:
13901:; Maass, Wolfgang; Turán, Gyorgy (1988), "On the Communication Complexity of Graph Properties",
8118:
parts. As with Szemerédi's regularity lemma, the weak regularity also induces a counting lemma.
4486:
4453:
14401:
14124:
11338:
9766:
8178:
8064:
7982:
7740:
7497:
12888:
12149:
6619:
4779:
3380:
14793:, Bolyai Soc. Math. Stud., vol. 2, János Bolyai Math. Soc., Budapest, pp. 295–352,
14221:
8200:
7394:
6168:
5340:
79:
72:-regularity. To understand what this means, we first state some definitions. In what follows
28:
14866:
14721:
14848:
14798:
14757:
14688:
14666:
14611:
14575:
14547:
14379:
14316:
14260:
14240:
14194:
14146:
14081:
14039:
13997:
13971:
13677:
13609:
13551:
13500:
13455:
13408:
13343:
13236:
12122:
12095:
12016:
11989:
11725:
11698:
11647:
11620:
11593:
11566:
9789:
6779:
6141:
5814:
5400:
5373:
5102:
4996:
4969:
3676:
2545:
2498:
940:
890:
43:
can be applied to dense graphs like counting the copies of a given subgraph within graphs.
14810:
14786:
13326:
Szemerédi's regularity lemma can be interpreted as saying that the space of all graphs is
9099:
8175:
The smaller bounds of the weak regularity lemma allow for efficient algorithms to find an
4872:
8:
13866:
Bansal, Nikhil; Williams, Ryan (2009), "Regularity Lemmas and
Combinatorial Algorithms",
13579:
10260:
8718:
We apply the regularity lemma repeatedly to prove the stronger version. A rough outline:
8228:
8220:
7330:
3354:
14670:
14615:
14579:
14244:
13681:
13210:, require many pairs of partitions (but a small fraction of all pairs) to be irregular.
14761:
14656:
14627:
14601:
14565:
14451:
14357:
14320:
14264:
14230:
14198:
14150:
14085:
13924:
13819:
13776:
13759:
13737:
13693:
13667:
13639:
13613:
13555:
13527:
13459:
13412:
13384:
13283:
12878:{\displaystyle P(\sum _{i,j}|d(W_{i},W_{j})-d(V_{i},V_{j})|^{2}\geq 8|P|^{2}r)\leq 1/2}
11294:
10687:
10418:
9040:
8320:
8300:
7933:
7804:
7080:
7060:
6866:
6846:
6759:
3970:
2627:
2525:
2478:
2458:
2438:
2418:
2398:
2154:
2134:
480:
460:
14814:
14345:
14290:
13985:
13945:
13941:
13815:
13342:; another version of the regularity lemma simply states that the space of graphons is
13167:
9459:
8215:
The strong regularity lemma is a stronger variation of the regularity lemma proven by
7848:
44:
14834:
14745:
14535:
14441:
14104:
14053:
14015:
13914:
13879:
13837:
13697:
13463:
13179:
13149:
14765:
14631:
14455:
14268:
14202:
14154:
13928:
13823:
13741:
13617:
13559:
859:
The boundaries of irregularity witnessing subsets refine each part of the partition.
14863:
14826:
14735:
14674:
14619:
14527:
14477:
14433:
14411:
14367:
14324:
14304:
14248:
14182:
14134:
14089:
14069:
14027:
13957:
13906:
13903:
Proceedings of the twentieth annual ACM symposium on Theory of computing - STOC '88
13871:
13849:
13811:
13780:
13768:
13729:
13685:
13597:
13537:
13488:
13443:
13416:
13394:
7349:
48:
13898:
14844:
14794:
14753:
14684:
14543:
14375:
14312:
14256:
14190:
14142:
14108:
14077:
14035:
13993:
13967:
13797:
13689:
13605:
13547:
13496:
13451:
13404:
13327:
1246:, we define the energy to be the sum of the energies between each pair of parts:
14252:
14011:
13754:
5289:
that witness irregularity (do this simultaneously for all irregular pairs). Let
1776:
Note that energy is between 0 and 1 because edge density is bounded above by 1:
266:
Subset pairs of a regular pair are similar in edge density to the original pair.
14216:
14168:
13715:
13542:
13492:
13429:
13183:
13153:
863:
We shall find an ε-regular partition for a given graph following an algorithm:
14740:
14186:
14171:(2006), "Quasirandomness, counting and regularity for 3-uniform hypergraphs",
13399:
9613:{\displaystyle q({\mathcal {P}}_{i+1})-q({\mathcal {P}}_{i})<\epsilon _{0}}
8195:-regular partition. Graph regularity has further been used in various area of
14875:
14789:(1996), "Szemerédi's regularity lemma and its applications in graph theory",
14749:
14539:
14437:
14295:
13720:
13588:
13583:
13171:
8224:
40:
14830:
14679:
14219:(2007), "Hypergraph regularity and the multidimensional Szemerédi theorem",
13962:
11093:
We'll only prove the weaker result where the second condition only requires
8061:
The weak regularity lemma for graphons states that every graphon has a weak
19:
14415:
13718:; Kannan, Ravi (1999), "Quick Approximation to Matrices and Applications",
13366:
13335:
8138:. It has been shown that approximating the max-cut problem beyond 16/17 is
7356:
14031:
13772:
13733:
13601:
14702:
14644:
14496:
13875:
13840:; Shapira, Asaf (2012), "Random sampling and approximation of MAX-CSPs",
13513:
13199:
13192:
13187:
13129:
Gowers's construction for the lower bound of Szemerédi's regularity lemma
8135:
8131:
6686:, and it's bounded above by 1. Then this process can be repeated at most
5084:{\displaystyle q({\mathcal {Q}})\geq q({\mathcal {P}})+\varepsilon ^{5}.}
56:
14372:
10.1002/(SICI)1098-2418(199805)12:3<297::AID-RSA5>3.3.CO;2-W
13946:"On sets of integers containing no k elements in arithmetic progression"
13910:
10415:
A partition is equitable if the sizes of any two sets differ by at most
7737:
The weak regularity lemma for graphs states that every graph has a weak
2062:
Now, we start by proving that energy does not decrease upon refinement.
14308:
13905:, vol. 26, Association for Computing Machinery, pp. 186–191,
13792:
13476:
13447:
13432:(1997), "Lower bounds of tower type for Szemerédi's uniformity lemma",
13207:
13145:
439:
14531:
14138:
14073:
13853:
13836:
Dellamonica, Domingos; Kalyanasundaram, Subrahmanyam; Martin, Daniel;
873:
Find the subsets which witness ε-irregularity for each irregular pair.
14817:(2002), "The regularity lemma and its applications in graph theory",
14805:
14661:
14623:
14570:
14518:; Shapira, Asaf (2008), "Every monotone graph property is testable",
14515:
14362:
13793:
13575:
13532:
13370:
8216:
8127:
3666:. Thus the refinement step in the algorithm doesn't lose any energy.
13835:
13178:
The first constructive version was provided by Alon, Duke, Lefmann,
8662:{\displaystyle q({\mathcal {Q}})<q({\mathcal {P}})+\epsilon _{0}}
7355:
Szemerédi's regularity lemma does not provide meaningful results in
7329:
This can be combined with Szemerédi's regularity lemma to prove the
3424:
is further partitioned, the new partition is called a refinement of
13672:
13644:
13144:) he proved the full lemma. Extensions of the regularity method to
3344:{\displaystyle q({\mathcal {P}}_{U},{\mathcal {P}}_{W})\geq q(U,W)}
2242:{\displaystyle q({\mathcal {P}}_{U},{\mathcal {P}}_{W})\geq q(U,W)}
855:
14606:
14482:
14235:
13868:
2009 50th Annual IEEE Symposium on
Foundations of Computer Science
13389:
13339:
10980:{\displaystyle |d(W_{i},W_{j})-d(V_{i},V_{j})|\leq \epsilon _{0}}
9449:{\displaystyle q({\mathcal {P}}_{i+1})\geq q({\mathcal {P}}_{i})}
8139:
7798:
745:
2069:(Energy is nondecreasing under partitioning) For any partitions
14724:; Szegedy, Balázs (2007), "Szemerédi's lemma for the analyst",
13053:{\displaystyle d(W_{i},W_{j})-d(V_{i},V_{j})\geq \epsilon _{0}}
9377:{\displaystyle \leq M(\epsilon _{|P_{i}|})|{\mathcal {P}}_{i}|}
8051:{\displaystyle \|W-W_{\mathcal {P}}\|_{\square }\leq \epsilon }
262:
9212:
9209:
14791:
Combinatorics, Paul Erdős is eighty, Vol. 2 (Keszthely, 1993)
13950:
Polska
Akademia Nauk. Instytut Matematyczny. Acta Arithmetica
13574:
434:-regular partition should be one where each pair of parts is
221:{\displaystyle d(X,Y):={\frac {\left|E(X,Y)\right|}{|X||Y|}}}
10547:{\displaystyle \epsilon _{0}\geq \epsilon _{1}\geq ...>0}
8290:{\displaystyle \epsilon _{0}\geq \epsilon _{1}\geq ...>0}
4443:{\displaystyle {\frac {|U_{1}|}{|U|}}{\frac {|W_{1}|}{|W|}}}
446:-regular partitions to be one where most pairs of parts are
419:{\displaystyle \left|d(X,Y)-d(A,B)\right|\leq \varepsilon .}
13060:. By union bound, the probability that all conditions hold
9125:
corresponds to the bound of parts in regularity lemma when
13373:(2012), "Bounds for graph regularity and removal lemmas",
2388:{\displaystyle {\mathcal {P}}_{W}=\{W_{1},\ldots ,W_{l}\}}
2319:{\displaystyle {\mathcal {P}}_{U}=\{U_{1},\ldots ,U_{k}\}}
1235:{\displaystyle {\mathcal {P}}_{W}=\{W_{1},\ldots ,W_{l}\}}
1162:{\displaystyle {\mathcal {P}}_{U}=\{U_{1},\ldots ,U_{k}\}}
23:
The edges between parts behave in a "random-like" fashion.
13800:(2003), "Random sampling and approximation of MAX-CSPs",
12092:, by union bound, the probability that at least one pair
11455:{\displaystyle q({\mathcal {Q}})-q({\mathcal {P}})\leq r}
11267:. We apply the strong regularity lemma to find equitable
9673:{\displaystyle ({\mathcal {P}}_{i},{\mathcal {P}}_{i+1})}
1090:{\displaystyle q(U,W):={\frac {|U||W|}{n^{2}}}d(U,W)^{2}}
837:
does indeed grow very fast and is at least as large as a
4344:{\displaystyle |Z-\mathbb {E} |=|d(U_{1},W_{1})-d(U,W)|}
3659:{\displaystyle q({\mathcal {P'}})\geq q({\mathcal {P}})}
14348:(1998), "An algorithmic version of the blow-up lemma",
7359:. Since sparse graphs have subconstant edge densities,
14819:
Theoretical aspects of computer science (Tehran, 2000)
14336:
14281:
13657:
10346:{\displaystyle M=\max _{i\leq 1/\epsilon _{0}+2}M_{i}}
7484:{\displaystyle {\mathcal {P}}=\{V_{1},\ldots ,V_{k}\}}
6299:
6100:
6043:
4862:{\displaystyle {\mathcal {P}}=\{V_{1},\ldots ,V_{k}\}}
4178:
4140:
3852:
3814:
3503:{\displaystyle {\mathcal {P}}=\{V_{1},\ldots ,V_{m}\}}
1441:{\displaystyle {\mathcal {P}}=\{V_{1},\ldots ,V_{k}\}}
878:
Refine the partition using all the witnessing subsets.
552:{\displaystyle {\mathcal {P}}=\{V_{1},\ldots ,V_{k}\}}
14825:, vol. 2292, Springer, Berlin, pp. 84–112,
13516:(2006), "A variant of the hypergraph removal lemma",
13309:
13286:
13266:
13239:
13219:
13066:
12969:
12922:
12891:
12722:
12185:
12152:
12125:
12098:
12046:
12019:
11992:
11968:
11816:
11792:
11754:
11728:
11701:
11677:
11650:
11623:
11596:
11569:
11511:
11468:
11410:
11386:
11341:
11317:
11297:
11273:
11227:
11186:
11145:
11099:
11044:
10993:
10886:
10847:
10806:
10760:
10713:
10690:
10650:
10586:
10560:
10502:
10465:
10441:
10421:
10359:
10289:
10263:
10181:
10026:
9976:
9922:
9873:
9819:
9792:
9769:
9729:
9686:
9626:
9541:
9494:
9462:
9390:
9310:
9279:
9235:
9198:
9160:
9131:
9102:
9063:
9043:
9016:
8985:
8957:
8933:
8893:
8856:
8815:
8791:
8759:
8728:
8676:
8610:
8567:
8543:
8516:
8492:
8466:
8442:
8418:
8394:
8367:
8343:
8323:
8303:
8245:
8181:
8148:
8087:
8067:
8010:
7985:
7959:
7936:
7912:
7883:
7851:
7827:
7807:
7763:
7743:
7555:
7520:
7500:
7435:
7397:
7365:
7181:
7165:{\displaystyle e(H)\varepsilon |V_{1}|\cdots |V_{k}|}
7103:
7083:
7063:
7016:
6996:
6950:
6889:
6869:
6849:
6823:
6782:
6762:
6722:
6692:
6665:
6645:
6622:
6602:
6578:
6551:
6340:
6266:
6228:
6204:
6171:
6144:
5871:
5844:
5817:
5779:
5433:
5403:
5376:
5343:
5319:
5295:
5249:
5203:
5183:
5137:
5105:
5029:
4999:
4972:
4948:
4924:
4904:
4875:
4813:
4782:
4525:
4489:
4456:
4357:
4244:
3996:
3973:
3786:
3731:
3711:
3679:
3615:
3591:
3562:
3516:
3454:
3430:
3406:
3383:
3357:
3279:
3217:
2925:
2653:
2630:
2575:
2548:
2528:
2501:
2481:
2461:
2441:
2421:
2401:
2332:
2263:
2177:
2157:
2137:
2106:
2075:
1785:
1528:
1482:
1458:
1392:
1255:
1179:
1106:
998:
943:
893:
591:
566:
503:
483:
463:
357:
136:
14708:
The spectral proof of the
Szemeredi regularity lemma
13195:'s blog and implicitly mentioned in various papers.
14391:
10250:{\displaystyle M_{i+1}=M(\epsilon _{|M_{i}|})M_{i}}
6324:{\displaystyle \{A^{i,j},V_{i}\backslash A^{i,j}\}}
14647:(2006), "Szemerédi's regularity lemma revisited",
14502:Szemeredi's regularity lemma via random partitions
13757:(2001), "Some Optimal Inapproximability Results",
13315:
13292:
13272:
13252:
13225:
13109:
13052:
12955:
12908:
12877:
12706:
12169:
12138:
12111:
12084:
12032:
12005:
11978:
11954:
11802:
11778:
11737:
11714:
11687:
11663:
11636:
11609:
11582:
11555:
11494:
11454:
11396:
11372:
11327:
11303:
11283:
11259:
11210:
11172:
11131:
11068:
11030:
10979:
10871:
10833:
10792:
10745:
10696:
10676:
10636:
10572:
10546:
10475:
10451:
10427:
10400:
10345:
10275:
10249:
10167:
10012:
9962:
9908:
9859:
9805:
9778:
9755:
9712:
9672:
9612:
9527:
9480:
9448:
9376:
9296:
9265:
9221:
9184:
9143:
9117:
9088:
9049:
9029:
9002:
8967:
8943:
8919:
8879:
8842:
8801:
8772:
8745:
8702:
8661:
8594:
8553:
8529:
8502:
8476:
8452:
8428:
8404:
8385:such that the following properties are satisfied:
8377:
8353:
8329:
8309:
8289:
8187:
8164:
8110:
8073:
8050:
7991:
7969:
7942:
7922:
7898:
7869:
7837:
7813:
7786:
7749:
7726:
7538:
7506:
7483:
7421:
7371:
7317:
7164:
7089:
7069:
7049:
7002:
6982:
6936:
6875:
6855:
6835:
6809:
6768:
6728:
6708:
6678:
6651:
6628:
6608:
6588:
6564:
6534:
6323:
6252:
6214:
6190:
6157:
6127:
5854:
5830:
5803:
5762:
5416:
5389:
5362:
5329:
5305:
5281:
5235:
5189:
5169:
5123:
5083:
5012:
4985:
4958:
4934:
4910:
4890:
4861:
4788:
4769:
4508:
4475:
4442:
4343:
4227:
3979:
3951:
3769:
3717:
3697:
3658:
3601:
3577:
3548:
3502:
3440:
3416:
3389:
3369:
3343:
3265:
3200:
2905:
2636:
2616:
2561:
2534:
2514:
2487:
2467:
2447:
2427:
2407:
2387:
2318:
2241:
2163:
2143:
2123:
2092:
2051:
1764:
1511:
1468:
1440:
1375:
1234:
1161:
1089:
965:
906:
718:
572:
551:
489:
469:
418:
246:denotes the set of edges having one end vertex in
220:
14862:Edmonds, Chelsea; Koutsoukou-Argyraki, Angeliki;
11862:
11849:
11031:{\displaystyle \epsilon _{0}|{\mathcal {P}}|^{2}}
7801:by defining a stepping operator. Given a graphon
3266:{\displaystyle \mathbb {E} \geq \mathbb {E} ^{2}}
14873:
14781:
14102:
13992:, vol. 260, Paris: CNRS, pp. 399–401,
13897:
10637:{\displaystyle {\mathcal {P}}={V_{1},...,V_{k}}}
10297:
8880:{\displaystyle {\mathcal {Q}}\leq \epsilon _{0}}
6937:{\displaystyle V_{1},\dots ,V_{k}\subseteq V(G)}
6716:times, before it terminates and we must have an
1512:{\displaystyle q({\mathcal {P}},{\mathcal {P}})}
13338:). Limits in this metric can be represented by
13213:It is a common variant in the definition of an
9713:{\displaystyle ({\mathcal {P}},{\mathcal {Q}})}
8920:{\displaystyle ({\mathcal {P}},{\mathcal {Q}})}
7333:. The graph removal lemma can be used to prove
6659:-regular. But in each step energy increases by
6616:-regular, so we deduce the desired inequality.
14056:; Skokan, Jozef (2004), "Regularity lemma for
13865:
10704:where the following properties are satisfied:
9963:{\displaystyle |{\mathcal {P}}_{0}|\leq M_{0}}
9860:{\displaystyle |{\mathcal {P}}_{i}|\leq M_{i}}
14720:
13632:Graph Compression Using The Regularity Method
13586:(2000), "Efficient testing of large graphs",
11962:vertex pairs live between irregular pairs in
5862:. By Lemma 1, the above quantity is at least
3770:{\displaystyle U_{1}\subset U,W_{1}\subset W}
14467:
14427:
14018:(2002), "Extremal problems on set systems",
13714:
11786:the first condition is trivially true since
11722:satisfy all the conditions with probability
11550:
11518:
11173:{\displaystyle \epsilon _{|{\mathcal {P}}|}}
10834:{\displaystyle \epsilon _{|{\mathcal {P}}|}}
8843:{\displaystyle \epsilon _{|{\mathcal {P}}|}}
8595:{\displaystyle \epsilon _{|{\mathcal {P}}|}}
8460:is the union of some collection of parts in
8033:
8011:
7478:
7446:
7204:
7192:
7029:
7017:
6318:
6267:
6119:
6068:
6062:
6011:
4856:
4824:
4191:
4159:
4153:
4121:
3865:
3833:
3827:
3795:
3497:
3465:
2382:
2350:
2313:
2281:
1435:
1403:
1229:
1197:
1156:
1124:
546:
514:
438:-regular. However, some graphs, such as the
14514:
13280:-fraction of the size of the vertex set of
8337:, we can obtain two (equitable) partitions
8121:
7382:
14052:
14010:
13796:; Fernandez de la Vega, W.; Kannan, Ravi;
13481:Journal of the London Mathematical Society
12085:{\displaystyle \leq {\frac {1}{3|P|^{2}}}}
8210:
14739:
14678:
14660:
14605:
14569:
14481:
14432:, IEEE Computer Society, pp. 12–20,
14405:
14361:
14234:
14128:
13984:
13961:
13940:
13671:
13643:
13541:
13531:
13398:
13388:
13365:
13141:
13133:
13120:
12596:
12465:
12318:
11810:is an equitable partition. Since at most
11556:{\displaystyle P=\{V_{1},\cdots ,V_{k}\}}
9384:parts. By the energy increment argument,
7335:Roth's Theorem on Arithmetic Progressions
4918:-regular, then there exists a refinement
4565:
4548:
4257:
4043:
4019:
3243:
3219:
2927:
2655:
14559:
14174:Combinatorics, Probability and Computing
14111:(2006), "The counting lemma for regular
13988:(1978), "Regular partitions of graphs",
13361:
13359:
13124:
7057:. Then, the number of labeled copies of
6331:. By lemma 2, the above sum is at least
4807:(Energy increment lemma) If a partition
854:
795:-regular partition of the vertex set of
261:
18:
14470:The Electronic Journal of Combinatorics
13803:Journal of Computer and System Sciences
13629:
13479:(1953), "On certain sets of integers",
11779:{\displaystyle \delta ={\frac {1}{2M}}}
10496:For any infinite sequence of constants
10410:
9528:{\displaystyle i\leq 1/\epsilon _{0}+1}
8239:For any infinite sequence of constants
7348:The graph removal lemma generalizes to
6744:
14874:
14591:
14562:A Simple Regularization of Hypergraphs
14215:
14167:
13753:
13428:
11495:{\displaystyle |{\mathcal {Q}}|\leq M}
11260:{\displaystyle r=\epsilon _{0}^{3}/20}
9909:{\displaystyle M_{0}=M(\epsilon _{0})}
8703:{\displaystyle |{\mathcal {Q}}|\leq M}
7906:as a step-graphon with steps given by
6253:{\displaystyle {\mathcal {Q}}_{V_{i}}}
5804:{\displaystyle {\mathcal {Q}}_{V_{i}}}
828:
14649:Contributions to Discrete Mathematics
13710:
13708:
13706:
13570:
13568:
13356:
11088:
10486:
9756:{\displaystyle {\mathcal {P}}_{i+1},}
9089:{\displaystyle \leq M(\epsilon _{0})}
870:While the partition isn't ε-regular:
13842:SIAM Journal on Discrete Mathematics
13829:
13475:
12956:{\displaystyle \epsilon _{0}|P|^{2}}
9222:{\displaystyle {\mathcal {P_{i+1}}}}
8850:regular. If the energy increment of
5282:{\displaystyle A^{j,i}\subset V_{j}}
5236:{\displaystyle A^{i,j}\subset V_{i}}
831:found examples of graphs for which
14701:
14643:
14495:
13786:
13512:
13469:
13110:{\displaystyle \geq 1-1/2-1/3>0}
11069:{\displaystyle 1\leq i\leq j\leq k}
10872:{\displaystyle 1\leq i\leq j\leq k}
10746:{\displaystyle |W_{i}|>\delta n}
10580:such that there exists a partition
9266:{\displaystyle \epsilon _{|P_{i}|}}
13:
14775:
14350:Random Structures & Algorithms
14117:Random Structures & Algorithms
14062:Random Structures & Algorithms
14020:Random Structures & Algorithms
13891:
13859:
13747:
13703:
13651:
13565:
12227:
12208:
11971:
11853:
11795:
11680:
11476:
11438:
11419:
11389:
11320:
11276:
11211:{\displaystyle 1\leq i<j\leq k}
11158:
11011:
10819:
10677:{\displaystyle W_{i}\subset V_{i}}
10589:
10468:
10444:
10100:
9931:
9828:
9770:
9733:
9702:
9692:
9650:
9633:
9583:
9551:
9432:
9400:
9358:
9297:{\displaystyle {\mathcal {P_{i}}}}
9287:
9283:
9206:
9202:
9003:{\displaystyle {\mathcal {P}}_{0}}
8989:
8960:
8936:
8909:
8899:
8859:
8828:
8794:
8746:{\displaystyle {\mathcal {P}}_{0}}
8732:
8684:
8638:
8619:
8580:
8546:
8495:
8469:
8445:
8421:
8397:
8370:
8346:
8111:{\displaystyle 4^{\epsilon ^{-2}}}
8026:
7962:
7915:
7890:
7830:
7787:{\displaystyle 4^{\epsilon ^{-2}}}
7438:
7337:, and a generalization of it, the
6581:
6232:
6207:
5847:
5783:
5739:
5715:
5633:
5609:
5527:
5503:
5442:
5322:
5298:
5057:
5038:
4951:
4927:
4816:
3648:
3625:
3594:
3566:
3457:
3433:
3409:
3306:
3289:
3184:
3167:
2569:. Then define the random variable
2336:
2267:
2204:
2187:
2124:{\displaystyle {\mathcal {P}}_{W}}
2110:
2093:{\displaystyle {\mathcal {P}}_{U}}
2079:
1794:
1537:
1501:
1491:
1461:
1395:
1282:
1265:
1183:
1110:
771:vertices, there exists an integer
506:
53:theorem on arithmetic progressions
14:
14898:
14856:
14823:Lecture Notes in Computer Science
14727:Geometric and Functional Analysis
13623:
13435:Geometric and Functional Analysis
13376:Geometric and Functional Analysis
10013:{\displaystyle |P_{i}|\leq M_{i}}
9456:. Since the energy is bounded in
7514:-regular if for any pair of sets
6836:{\displaystyle \varepsilon >0}
6709:{\displaystyle \varepsilon ^{-5}}
3556:proves that for every refinement
13636:Ca' Foscari University of Venice
11986:, the probability that the pair
11311:regular partition and equitable
8081:-regular partition into at most
7899:{\displaystyle W_{\mathcal {P}}}
7757:-regular partition into at most
6679:{\displaystyle \varepsilon ^{5}}
6565:{\displaystyle \varepsilon ^{5}}
3510:, applying Lemma 1 to each pair
2617:{\displaystyle Z=d(U_{i},W_{j})}
882:We apply a technique called the
16:Concept in extremal graph theory
14714:
14695:
14637:
14585:
14553:
14508:
14489:
14461:
14421:
14385:
14330:
14275:
14209:
14161:
14096:
14046:
14004:
13978:
13934:
13519:Journal of Combinatorial Theory
8785:Repeatedly find its refinement
7797:This notion can be extended to
7050:{\displaystyle \{i,j\}\in E(H)}
6883:-vertex graph with vertex sets
6739:
6545:But the second sum is at least
3578:{\displaystyle {\mathcal {P'}}}
2624:. Let us look at properties of
841:-level iterated exponential of
821:-level iterated exponential of
13506:
13422:
13170:later (in 1997) proved in the
13034:
13008:
12999:
12973:
12943:
12934:
12858:
12845:
12836:
12819:
12814:
12788:
12779:
12753:
12746:
12726:
12690:
12685:
12659:
12650:
12624:
12617:
12582:
12573:
12543:
12538:
12512:
12503:
12477:
12470:
12451:
12442:
12396:
12391:
12365:
12356:
12330:
12323:
12300:
12285:
12280:
12265:
12232:
12222:
12213:
12203:
12069:
12060:
11979:{\displaystyle {\mathcal {Q}}}
11939:
11930:
11920:
11905:
11900:
11885:
11833:
11824:
11803:{\displaystyle {\mathcal {Q}}}
11688:{\displaystyle {\mathcal {Q}}}
11482:
11470:
11443:
11433:
11424:
11414:
11397:{\displaystyle {\mathcal {P}}}
11360:
11351:
11328:{\displaystyle {\mathcal {Q}}}
11284:{\displaystyle {\mathcal {P}}}
11164:
11152:
11126:
11100:
11018:
11005:
10960:
10956:
10930:
10921:
10895:
10888:
10825:
10813:
10787:
10761:
10730:
10715:
10491:
10476:{\displaystyle {\mathcal {Q}}}
10452:{\displaystyle {\mathcal {P}}}
10401:{\displaystyle |P|,|Q|\leq M.}
10385:
10377:
10369:
10361:
10257:and the statement is true for
10234:
10228:
10213:
10204:
10152:
10146:
10131:
10122:
10112:
10093:
10089:
10083:
10068:
10059:
10049:
10028:
9993:
9978:
9943:
9924:
9903:
9890:
9840:
9821:
9707:
9687:
9667:
9627:
9594:
9577:
9568:
9545:
9475:
9463:
9443:
9426:
9417:
9394:
9370:
9351:
9347:
9341:
9326:
9317:
9257:
9242:
9112:
9106:
9083:
9070:
8968:{\displaystyle {\mathcal {Q}}}
8944:{\displaystyle {\mathcal {P}}}
8914:
8894:
8834:
8822:
8802:{\displaystyle {\mathcal {Q}}}
8690:
8678:
8643:
8633:
8624:
8614:
8586:
8574:
8554:{\displaystyle {\mathcal {Q}}}
8503:{\displaystyle {\mathcal {P}}}
8477:{\displaystyle {\mathcal {Q}}}
8453:{\displaystyle {\mathcal {P}}}
8429:{\displaystyle {\mathcal {P}}}
8405:{\displaystyle {\mathcal {Q}}}
8378:{\displaystyle {\mathcal {Q}}}
8354:{\displaystyle {\mathcal {P}}}
8234:
8165:{\displaystyle \epsilon n^{2}}
7970:{\displaystyle {\mathcal {P}}}
7930:and values given by averaging
7923:{\displaystyle {\mathcal {P}}}
7864:
7852:
7838:{\displaystyle {\mathcal {P}}}
7714:
7705:
7689:
7668:
7663:
7642:
7638:
7612:
7576:
7564:
7539:{\displaystyle S,T\subseteq V}
7429:, a partition of its vertices
7416:
7404:
7303:
7288:
7253:
7227:
7219:
7213:
7158:
7143:
7135:
7120:
7113:
7107:
7044:
7038:
6977:
6951:
6931:
6925:
6804:
6798:
6792:
6786:
6589:{\displaystyle {\mathcal {P}}}
6515:
6500:
6495:
6480:
6448:
6422:
6411:
6385:
6371:
6364:
6358:
6346:
6215:{\displaystyle {\mathcal {Q}}}
6122:
6008:
5987:
5961:
5950:
5924:
5903:
5877:
5855:{\displaystyle {\mathcal {Q}}}
5757:
5709:
5688:
5662:
5651:
5603:
5582:
5556:
5545:
5497:
5483:
5476:
5470:
5458:
5447:
5437:
5330:{\displaystyle {\mathcal {P}}}
5306:{\displaystyle {\mathcal {Q}}}
5164:
5138:
5118:
5106:
5062:
5052:
5043:
5033:
4959:{\displaystyle {\mathcal {P}}}
4935:{\displaystyle {\mathcal {Q}}}
4885:
4879:
4733:
4729:
4717:
4708:
4682:
4676:
4669:
4661:
4654:
4639:
4628:
4620:
4613:
4598:
4588:
4579:
4575:
4569:
4555:
4552:
4541:
4535:
4433:
4425:
4418:
4403:
4392:
4384:
4377:
4362:
4337:
4333:
4321:
4312:
4286:
4279:
4271:
4267:
4261:
4246:
4217:
4205:
4101:
4093:
4088:
4080:
4054:
4047:
4036:
4023:
4012:
4006:
3932:
3924:
3919:
3911:
3891:
3879:
3692:
3680:
3653:
3643:
3634:
3619:
3602:{\displaystyle {\mathcal {P}}}
3543:
3517:
3441:{\displaystyle {\mathcal {P}}}
3417:{\displaystyle {\mathcal {P}}}
3338:
3326:
3317:
3283:
3254:
3247:
3236:
3223:
3195:
3161:
3151:
3143:
3138:
3130:
3104:
3077:
3067:
3059:
3052:
3037:
3026:
3018:
3011:
2996:
2944:
2931:
2900:
2888:
2875:
2867:
2862:
2854:
2848:
2836:
2824:
2798:
2788:
2780:
2773:
2758:
2747:
2739:
2732:
2717:
2665:
2659:
2611:
2585:
2236:
2224:
2215:
2181:
2026:
2011:
2006:
1991:
1933:
1906:
1886:
1871:
1866:
1851:
1799:
1789:
1753:
1726:
1706:
1691:
1686:
1671:
1619:
1593:
1542:
1532:
1506:
1486:
1469:{\displaystyle {\mathcal {P}}}
1370:
1344:
1293:
1259:
1078:
1065:
1045:
1037:
1032:
1024:
1014:
1002:
953:
945:
706:
701:
695:
688:
677:
662:
657:
642:
623:
597:
399:
387:
378:
366:
211:
203:
198:
190:
180:
168:
152:
140:
1:
13816:10.1016/S0022-0000(03)00008-4
13349:
11132:{\displaystyle (W_{i},W_{j})}
11079:
10793:{\displaystyle (W_{i},W_{j})}
9185:{\displaystyle i=0,1,\cdots }
9030:{\displaystyle \epsilon _{0}}
8773:{\displaystyle \epsilon _{0}}
8530:{\displaystyle \epsilon _{0}}
6983:{\displaystyle (V_{i},V_{j})}
5170:{\displaystyle (V_{i},V_{j})}
3549:{\displaystyle (V_{i},V_{j})}
735:Szemerédi's Regularity Lemma.
430:The natural way to define an
14560:Ishigami, Yoshiyasu (2006),
13690:10.1016/j.patcog.2019.107070
13334:) in a suitable metric (the
13316:{\displaystyle \varepsilon }
13273:{\displaystyle \varepsilon }
13226:{\displaystyle \varepsilon }
11695:. We argue that the subsets
11644:to be the set that contains
11563:, we randomly pick a vertex
10573:{\displaystyle \delta >0}
8197:theoretical computer science
7372:{\displaystyle \varepsilon }
7003:{\displaystyle \varepsilon }
6729:{\displaystyle \varepsilon }
6652:{\displaystyle \varepsilon }
6609:{\displaystyle \varepsilon }
5397:is partitioned into at most
5190:{\displaystyle \varepsilon }
4993:is partitioned into at most
4911:{\displaystyle \varepsilon }
3718:{\displaystyle \varepsilon }
730:Now we can state the lemma:
573:{\displaystyle \varepsilon }
62:
33:Szemerédi’s regularity lemma
7:
14253:10.4007/annals.2007.166.897
13630:Pelosin, Francesco (2018),
10483:are equitable partitions.
9144:{\displaystyle \epsilon =t}
8229:induced graph removal lemma
3273:. Rearranging, we get that
10:
14903:
13543:10.1016/j.jcta.2005.11.006
13182:and Yuster. Subsequently,
13152:and his collaborators and
12716:So by Markov's inequality
8713:
8297:, there exists an integer
5313:be a common refinement of
4509:{\displaystyle y\in W_{1}}
4476:{\displaystyle x\in U_{1}}
14741:10.1007/s00039-007-0599-6
14293:(1997), "Blow-up lemma",
14187:10.1017/S0963548305007236
13400:10.1007/s00039-012-0171-x
13260:whose size is at most an
11373:{\displaystyle r/|P|^{4}}
9779:{\displaystyle \forall i}
8188:{\displaystyle \epsilon }
8074:{\displaystyle \epsilon }
7992:{\displaystyle \epsilon }
7750:{\displaystyle \epsilon }
7507:{\displaystyle \epsilon }
3725:-regular as witnessed by
1386:Finally, for a partition
884:energy increment argument
765:is a graph with at least
14438:10.1109/SFCS.1996.548459
13493:10.1112/jlms/s1-28.1.104
12909:{\displaystyle \geq 1/2}
12170:{\displaystyle \leq 1/3}
8927:. Otherwise, we replace
8436:, that is every part of
8317:such that for any graph
8205:communication complexity
8122:Algorithmic applications
7383:Frieze-Kannan regularity
7339:hypergraph removal lemma
6629:{\displaystyle \square }
5424:parts as desired. Then,
4789:{\displaystyle \square }
3673:(Energy boost lemma) If
3390:{\displaystyle \square }
850:
753:there exists an integer
285:, a pair of vertex sets
270:We call a pair of parts
14831:10.1007/3-540-45878-6_3
14680:10.11575/cdm.v1i1.61900
14115:-uniform hypergraphs",
14060:-uniform hypergraphs",
13963:10.4064/aa-27-1-199-245
10841:-regular for each pair
8211:Strong regularity lemma
7422:{\displaystyle G=(V,E)}
7341:, can be used to prove
6191:{\displaystyle A^{i,j}}
5363:{\displaystyle A^{i,j}}
1452:, define the energy of
105:be disjoint subsets of
14882:Lemmas in graph theory
14416:10.1006/jagm.1994.1005
13317:
13294:
13274:
13254:
13227:
13130:
13121:History and extensions
13111:
13054:
12957:
12910:
12885:, so with probability
12879:
12708:
12171:
12140:
12113:
12086:
12034:
12007:
11980:
11956:
11804:
11780:
11739:
11716:
11689:
11665:
11638:
11611:
11584:
11557:
11496:
11456:
11398:
11380:regular refinement of
11374:
11329:
11305:
11285:
11261:
11212:
11174:
11133:
11070:
11032:
10981:
10873:
10835:
10794:
10747:
10698:
10678:
10638:
10574:
10548:
10477:
10453:
10429:
10402:
10347:
10277:
10251:
10169:
10014:
9964:
9910:
9861:
9807:
9780:
9757:
9714:
9674:
9614:
9529:
9482:
9450:
9378:
9298:
9273:regular refinement of
9267:
9223:
9186:
9145:
9119:
9090:
9051:
9031:
9004:
8969:
8945:
8921:
8881:
8844:
8803:
8774:
8747:
8704:
8663:
8596:
8555:
8531:
8504:
8478:
8454:
8430:
8406:
8379:
8355:
8331:
8311:
8291:
8189:
8166:
8112:
8075:
8052:
7993:
7971:
7944:
7924:
7900:
7871:
7839:
7815:
7788:
7751:
7728:
7608:
7540:
7508:
7485:
7423:
7373:
7327:
7319:
7286:
7166:
7091:
7071:
7051:
7004:
6984:
6938:
6877:
6857:
6837:
6811:
6770:
6730:
6710:
6680:
6653:
6637:
6630:
6610:
6590:
6566:
6536:
6325:
6254:
6216:
6192:
6159:
6129:
5856:
5832:
5805:
5764:
5418:
5391:
5364:
5331:
5307:
5283:
5237:
5191:
5171:
5125:
5093:
5085:
5014:
4987:
4960:
4936:
4912:
4892:
4863:
4798:
4790:
4771:
4510:
4477:
4444:
4345:
4229:
3981:
3961:
3953:
3771:
3719:
3699:
3660:
3603:
3579:
3550:
3504:
3442:
3418:
3398:
3391:
3371:
3345:
3267:
3202:
2991:
2970:
2907:
2712:
2691:
2638:
2618:
2563:
2536:
2516:
2489:
2469:
2449:
2429:
2409:
2389:
2320:
2251:
2243:
2165:
2145:
2125:
2094:
2053:
1986:
1965:
1846:
1825:
1774:
1766:
1666:
1645:
1589:
1568:
1513:
1470:
1442:
1377:
1340:
1319:
1236:
1163:
1091:
967:
908:
867:Start with a partition
860:
809:
728:
720:
574:
553:
491:
471:
428:
420:
267:
260:
222:
47:proved the lemma over
24:
14809:; Shokoufandeh, Ali;
14394:Journal of Algorithms
14222:Annals of Mathematics
14032:10.1002/rsa.10017.abs
13773:10.1145/502090.502098
13734:10.1007/s004930050052
13602:10.1007/s004930070001
13318:
13295:
13275:
13255:
13253:{\displaystyle V_{0}}
13228:
13128:
13112:
13055:
12958:
12911:
12880:
12709:
12172:
12141:
12139:{\displaystyle W_{i}}
12114:
12112:{\displaystyle W_{i}}
12087:
12035:
12033:{\displaystyle W_{j}}
12008:
12006:{\displaystyle W_{i}}
11981:
11957:
11805:
11781:
11740:
11738:{\displaystyle >0}
11717:
11715:{\displaystyle W_{i}}
11690:
11666:
11664:{\displaystyle v_{i}}
11639:
11637:{\displaystyle W_{i}}
11612:
11610:{\displaystyle V_{i}}
11585:
11583:{\displaystyle v_{i}}
11558:
11497:
11457:
11399:
11375:
11330:
11306:
11286:
11262:
11213:
11175:
11134:
11071:
11033:
10982:
10874:
10836:
10795:
10748:
10699:
10679:
10639:
10575:
10549:
10478:
10454:
10430:
10403:
10348:
10278:
10252:
10170:
10015:
9965:
9911:
9862:
9808:
9806:{\displaystyle M_{i}}
9781:
9758:
9715:
9675:
9615:
9530:
9488:, there must be some
9483:
9451:
9379:
9299:
9268:
9224:
9187:
9146:
9120:
9091:
9052:
9037:regular partition of
9032:
9005:
8970:
8946:
8922:
8882:
8845:
8804:
8775:
8748:
8705:
8664:
8597:
8556:
8532:
8505:
8479:
8455:
8431:
8407:
8380:
8356:
8332:
8312:
8292:
8201:matrix multiplication
8190:
8167:
8113:
8076:
8053:
7994:
7972:
7945:
7925:
7901:
7872:
7840:
7816:
7789:
7752:
7729:
7582:
7541:
7509:
7486:
7424:
7374:
7320:
7266:
7167:
7092:
7072:
7052:
7005:
6985:
6939:
6878:
6858:
6838:
6812:
6810:{\displaystyle V(H)=}
6771:
6754:Graph Counting Lemma.
6751:
6731:
6711:
6681:
6654:
6631:
6611:
6591:
6567:
6537:
6326:
6255:
6217:
6193:
6160:
6158:{\displaystyle V_{i}}
6130:
5857:
5833:
5831:{\displaystyle V_{i}}
5806:
5765:
5419:
5417:{\displaystyle 2^{k}}
5392:
5390:{\displaystyle V_{i}}
5365:
5332:
5308:
5284:
5238:
5192:
5172:
5126:
5124:{\displaystyle (i,j)}
5094:
5086:
5015:
5013:{\displaystyle 2^{k}}
4988:
4986:{\displaystyle V_{i}}
4961:
4937:
4913:
4893:
4864:
4802:
4791:
4772:
4511:
4478:
4445:
4346:
4230:
3982:
3962:
3954:
3772:
3720:
3700:
3698:{\displaystyle (U,W)}
3668:
3661:
3604:
3580:
3551:
3505:
3443:
3419:
3392:
3372:
3346:
3268:
3203:
2971:
2950:
2916:The second moment is
2908:
2692:
2671:
2644:. The expectation is
2639:
2619:
2564:
2562:{\displaystyle W_{j}}
2537:
2517:
2515:{\displaystyle U_{i}}
2490:
2470:
2450:
2430:
2410:
2390:
2321:
2252:
2244:
2166:
2146:
2126:
2095:
2064:
2054:
1966:
1945:
1826:
1805:
1767:
1646:
1625:
1569:
1548:
1514:
1471:
1443:
1378:
1320:
1299:
1237:
1164:
1092:
968:
966:{\displaystyle |V|=n}
916:
909:
907:{\displaystyle L^{2}}
858:
732:
721:
575:
554:
492:
472:
452:
421:
304:, if for all subsets
276:
265:
223:
90:
29:extremal graph theory
22:
14864:Paulson, Lawrence C.
13876:10.1109/FOCS.2009.76
13870:, pp. 745–754,
13580:Krivelevich, Michael
13307:
13284:
13264:
13237:
13217:
13064:
12967:
12920:
12889:
12720:
12183:
12150:
12123:
12096:
12044:
12017:
11990:
11966:
11814:
11790:
11752:
11726:
11699:
11675:
11648:
11621:
11594:
11567:
11509:
11466:
11408:
11384:
11339:
11315:
11295:
11271:
11225:
11184:
11143:
11097:
11042:
10991:
10884:
10845:
10804:
10758:
10711:
10688:
10648:
10584:
10558:
10500:
10463:
10439:
10419:
10411:Remarks on equitable
10357:
10287:
10261:
10179:
10024:
9974:
9920:
9871:
9817:
9790:
9767:
9727:
9684:
9624:
9539:
9492:
9460:
9388:
9308:
9277:
9233:
9196:
9158:
9129:
9118:{\displaystyle M(t)}
9100:
9061:
9041:
9014:
8983:
8955:
8931:
8891:
8854:
8813:
8789:
8757:
8726:
8674:
8608:
8565:
8541:
8514:
8490:
8464:
8440:
8416:
8392:
8365:
8341:
8321:
8301:
8243:
8179:
8146:
8085:
8065:
8008:
7983:
7957:
7934:
7910:
7881:
7849:
7825:
7805:
7761:
7741:
7553:
7518:
7498:
7433:
7395:
7363:
7179:
7101:
7081:
7061:
7014:
6994:
6948:
6887:
6867:
6847:
6821:
6780:
6760:
6745:Graph counting lemma
6736:-regular partition.
6720:
6690:
6663:
6643:
6620:
6600:
6576:
6549:
6338:
6264:
6226:
6202:
6169:
6142:
5869:
5842:
5815:
5811:is the partition of
5777:
5431:
5401:
5374:
5341:
5317:
5293:
5247:
5201:
5181:
5135:
5103:
5027:
4997:
4970:
4946:
4922:
4902:
4891:{\displaystyle V(G)}
4873:
4811:
4780:
4523:
4487:
4454:
4355:
4242:
3994:
3971:
3784:
3729:
3709:
3677:
3613:
3589:
3560:
3514:
3452:
3428:
3404:
3381:
3355:
3277:
3215:
2923:
2651:
2628:
2573:
2546:
2526:
2499:
2479:
2459:
2439:
2419:
2399:
2330:
2261:
2175:
2155:
2135:
2104:
2073:
1783:
1526:
1480:
1456:
1390:
1253:
1177:
1104:
996:
941:
891:
589:
564:
501:
481:
461:
355:
134:
14671:2005math......4472T
14616:2008arXiv0801.1698A
14594:Probability Surveys
14580:2006math.....12838I
14245:2007arXiv0710.3032G
13911:10.1145/62212.62228
13682:2020PatRe..9807070F
13660:Pattern Recognition
13138:Szemerédi's theorem
11248:
10276:{\displaystyle i+1}
8887:, we simply return
8130:for estimating the
7343:Szemerédi's theorem
7331:Graph removal lemma
6260:is a refinement of
3370:{\displaystyle U,W}
786: ≤
782: ≤
14887:Information theory
14811:Simonovits, Miklós
14476:(1), Article R17,
14309:10.1007/BF01196135
13760:Journal of the ACM
13578:; Fischer, Eldar;
13448:10.1007/PL00001621
13313:
13290:
13270:
13250:
13223:
13131:
13107:
13050:
12953:
12906:
12875:
12744:
12704:
12702:
12615:
12431:
12260:
12167:
12136:
12109:
12082:
12030:
12003:
11976:
11952:
11800:
11776:
11735:
11712:
11685:
11661:
11634:
11607:
11580:
11553:
11492:
11452:
11394:
11370:
11325:
11301:
11281:
11257:
11234:
11208:
11170:
11129:
11089:Proof of corollary
11066:
11028:
10977:
10869:
10831:
10790:
10743:
10694:
10674:
10634:
10570:
10544:
10487:A useful corollary
10473:
10449:
10425:
10398:
10343:
10332:
10273:
10247:
10175:, so we could set
10165:
10010:
9960:
9906:
9857:
9803:
9776:
9753:
9710:
9670:
9610:
9525:
9478:
9446:
9374:
9294:
9263:
9219:
9182:
9141:
9115:
9086:
9047:
9027:
9000:
8965:
8941:
8917:
8877:
8840:
8799:
8770:
8743:
8700:
8659:
8592:
8551:
8527:
8500:
8474:
8450:
8426:
8402:
8375:
8351:
8327:
8307:
8287:
8185:
8162:
8108:
8071:
8048:
7989:
7967:
7940:
7920:
7896:
7867:
7835:
7811:
7784:
7747:
7724:
7536:
7504:
7481:
7419:
7369:
7315:
7223:
7162:
7087:
7067:
7047:
7010:-regular whenever
7000:
6980:
6934:
6873:
6853:
6833:
6807:
6766:
6726:
6706:
6676:
6649:
6626:
6606:
6586:
6562:
6532:
6465:
6381:
6321:
6250:
6212:
6188:
6155:
6125:
6004:
5920:
5852:
5828:
5801:
5760:
5705:
5599:
5493:
5414:
5387:
5360:
5327:
5303:
5279:
5233:
5187:
5167:
5121:
5081:
5010:
4983:
4956:
4932:
4908:
4888:
4859:
4786:
4767:
4506:
4473:
4450:(corresponding to
4440:
4341:
4225:
3977:
3949:
3767:
3715:
3695:
3656:
3599:
3575:
3546:
3500:
3438:
3414:
3387:
3367:
3341:
3263:
3198:
2903:
2634:
2614:
2559:
2532:
2512:
2485:
2465:
2445:
2425:
2405:
2395:. Choose vertices
2385:
2316:
2239:
2161:
2141:
2121:
2090:
2049:
1762:
1509:
1466:
1438:
1373:
1232:
1159:
1087:
963:
904:
861:
739:ε > 0
716:
640:
580:-regular partition
570:
549:
487:
467:
416:
283:ε > 0
268:
218:
25:
14840:978-3-540-43328-6
14532:10.1137/050633445
14342:Sárközy, Gábor N.
14287:Sárközy, Gábor N.
14225:, Second Series,
14139:10.1002/rsa.20117
14074:10.1002/rsa.20017
13885:978-1-4244-5116-6
13854:10.1137/110846373
13293:{\displaystyle G}
13198:An inequality of
13148:were obtained by
12963:pairs could have
12729:
12600:
12593:
12462:
12416:
12315:
12245:
12080:
11950:
11860:
11844:
11774:
11304:{\displaystyle r}
10697:{\displaystyle i}
10428:{\displaystyle 1}
10296:
9970:. Note that when
9723:By our choice of
9050:{\displaystyle G}
8780:regular partition
8330:{\displaystyle G}
8310:{\displaystyle M}
7943:{\displaystyle W}
7814:{\displaystyle W}
7350:induced subgraphs
7187:
7090:{\displaystyle G}
7070:{\displaystyle H}
6876:{\displaystyle n}
6856:{\displaystyle G}
6769:{\displaystyle H}
6530:
6462:
6454:
6417:
6341:
6001:
5993:
5956:
5917:
5909:
5872:
5702:
5694:
5657:
5596:
5588:
5551:
5453:
4674:
4633:
4438:
4397:
4351:with probability
4238:But observe that
4106:
3980:{\displaystyle Z}
3947:
3156:
3072:
3031:
2880:
2793:
2752:
2637:{\displaystyle Z}
2535:{\displaystyle y}
2488:{\displaystyle x}
2468:{\displaystyle W}
2448:{\displaystyle y}
2428:{\displaystyle U}
2408:{\displaystyle x}
2164:{\displaystyle W}
2144:{\displaystyle U}
2041:
1901:
1721:
1060:
637:
629:
592:
490:{\displaystyle k}
470:{\displaystyle V}
216:
14894:
14851:
14815:Szemerédi, Endre
14801:
14769:
14768:
14743:
14718:
14712:
14711:
14699:
14693:
14691:
14682:
14664:
14641:
14635:
14634:
14624:10.1214/08-PS124
14609:
14589:
14583:
14582:
14573:
14557:
14551:
14550:
14512:
14506:
14505:
14493:
14487:
14486:
14485:
14465:
14459:
14458:
14425:
14419:
14418:
14409:
14389:
14383:
14382:
14365:
14346:Szemerédi, Endre
14334:
14328:
14327:
14291:Szemerédi, Endre
14279:
14273:
14271:
14238:
14213:
14207:
14205:
14181:(1–2): 143–184,
14165:
14159:
14157:
14132:
14114:
14109:Schacht, Mathias
14103:Nagle, Brendan;
14100:
14094:
14092:
14059:
14050:
14044:
14042:
14008:
14002:
14000:
13986:Szemerédi, Endre
13982:
13976:
13974:
13965:
13942:Szemerédi, Endre
13938:
13932:
13931:
13895:
13889:
13888:
13863:
13857:
13856:
13833:
13827:
13826:
13798:Karpinksi, Marek
13790:
13784:
13783:
13751:
13745:
13744:
13712:
13701:
13700:
13675:
13655:
13649:
13648:
13647:
13627:
13621:
13620:
13572:
13563:
13562:
13545:
13535:
13526:(7): 1257–1280,
13510:
13504:
13503:
13473:
13467:
13466:
13426:
13420:
13419:
13402:
13392:
13383:(5): 1191–1256,
13363:
13322:
13320:
13319:
13314:
13299:
13297:
13296:
13291:
13279:
13277:
13276:
13271:
13259:
13257:
13256:
13251:
13249:
13248:
13232:
13230:
13229:
13224:
13134:Szemerédi (1975)
13116:
13114:
13113:
13108:
13097:
13083:
13059:
13057:
13056:
13051:
13049:
13048:
13033:
13032:
13020:
13019:
12998:
12997:
12985:
12984:
12962:
12960:
12959:
12954:
12952:
12951:
12946:
12937:
12932:
12931:
12915:
12913:
12912:
12907:
12902:
12884:
12882:
12881:
12876:
12871:
12854:
12853:
12848:
12839:
12828:
12827:
12822:
12813:
12812:
12800:
12799:
12778:
12777:
12765:
12764:
12749:
12743:
12713:
12711:
12710:
12705:
12703:
12699:
12698:
12693:
12684:
12683:
12671:
12670:
12649:
12648:
12636:
12635:
12620:
12614:
12599:
12594:
12592:
12591:
12590:
12585:
12576:
12564:
12556:
12552:
12551:
12546:
12537:
12536:
12524:
12523:
12502:
12501:
12489:
12488:
12473:
12468:
12463:
12461:
12460:
12459:
12454:
12445:
12433:
12430:
12409:
12405:
12404:
12399:
12390:
12389:
12377:
12376:
12355:
12354:
12342:
12341:
12326:
12321:
12316:
12314:
12313:
12304:
12303:
12298:
12297:
12288:
12283:
12278:
12277:
12268:
12262:
12259:
12238:
12231:
12230:
12212:
12211:
12176:
12174:
12173:
12168:
12163:
12145:
12143:
12142:
12137:
12135:
12134:
12118:
12116:
12115:
12110:
12108:
12107:
12091:
12089:
12088:
12083:
12081:
12079:
12078:
12077:
12072:
12063:
12051:
12039:
12037:
12036:
12031:
12029:
12028:
12012:
12010:
12009:
12004:
12002:
12001:
11985:
11983:
11982:
11977:
11975:
11974:
11961:
11959:
11958:
11953:
11951:
11949:
11948:
11947:
11942:
11933:
11924:
11923:
11918:
11917:
11908:
11903:
11898:
11897:
11888:
11882:
11880:
11879:
11867:
11866:
11865:
11852:
11845:
11843:
11842:
11841:
11836:
11827:
11818:
11809:
11807:
11806:
11801:
11799:
11798:
11785:
11783:
11782:
11777:
11775:
11773:
11762:
11744:
11742:
11741:
11736:
11721:
11719:
11718:
11713:
11711:
11710:
11694:
11692:
11691:
11686:
11684:
11683:
11670:
11668:
11667:
11662:
11660:
11659:
11643:
11641:
11640:
11635:
11633:
11632:
11616:
11614:
11613:
11608:
11606:
11605:
11589:
11587:
11586:
11581:
11579:
11578:
11562:
11560:
11559:
11554:
11549:
11548:
11530:
11529:
11505:Now assume that
11501:
11499:
11498:
11493:
11485:
11480:
11479:
11473:
11461:
11459:
11458:
11453:
11442:
11441:
11423:
11422:
11403:
11401:
11400:
11395:
11393:
11392:
11379:
11377:
11376:
11371:
11369:
11368:
11363:
11354:
11349:
11334:
11332:
11331:
11326:
11324:
11323:
11310:
11308:
11307:
11302:
11290:
11288:
11287:
11282:
11280:
11279:
11266:
11264:
11263:
11258:
11253:
11247:
11242:
11217:
11215:
11214:
11209:
11179:
11177:
11176:
11171:
11169:
11168:
11167:
11162:
11161:
11155:
11138:
11136:
11135:
11130:
11125:
11124:
11112:
11111:
11075:
11073:
11072:
11067:
11037:
11035:
11034:
11029:
11027:
11026:
11021:
11015:
11014:
11008:
11003:
11002:
10986:
10984:
10983:
10978:
10976:
10975:
10963:
10955:
10954:
10942:
10941:
10920:
10919:
10907:
10906:
10891:
10878:
10876:
10875:
10870:
10840:
10838:
10837:
10832:
10830:
10829:
10828:
10823:
10822:
10816:
10799:
10797:
10796:
10791:
10786:
10785:
10773:
10772:
10752:
10750:
10749:
10744:
10733:
10728:
10727:
10718:
10703:
10701:
10700:
10695:
10683:
10681:
10680:
10675:
10673:
10672:
10660:
10659:
10643:
10641:
10640:
10635:
10633:
10632:
10631:
10607:
10606:
10593:
10592:
10579:
10577:
10576:
10571:
10553:
10551:
10550:
10545:
10525:
10524:
10512:
10511:
10482:
10480:
10479:
10474:
10472:
10471:
10458:
10456:
10455:
10450:
10448:
10447:
10434:
10432:
10431:
10426:
10407:
10405:
10404:
10399:
10388:
10380:
10372:
10364:
10352:
10350:
10349:
10344:
10342:
10341:
10331:
10324:
10323:
10314:
10282:
10280:
10279:
10274:
10256:
10254:
10253:
10248:
10246:
10245:
10233:
10232:
10231:
10226:
10225:
10216:
10197:
10196:
10174:
10172:
10171:
10166:
10164:
10163:
10151:
10150:
10149:
10144:
10143:
10134:
10115:
10110:
10109:
10104:
10103:
10096:
10088:
10087:
10086:
10081:
10080:
10071:
10052:
10047:
10046:
10031:
10019:
10017:
10016:
10011:
10009:
10008:
9996:
9991:
9990:
9981:
9969:
9967:
9966:
9961:
9959:
9958:
9946:
9941:
9940:
9935:
9934:
9927:
9915:
9913:
9912:
9907:
9902:
9901:
9883:
9882:
9866:
9864:
9863:
9858:
9856:
9855:
9843:
9838:
9837:
9832:
9831:
9824:
9812:
9810:
9809:
9804:
9802:
9801:
9785:
9783:
9782:
9777:
9762:
9760:
9759:
9754:
9749:
9748:
9737:
9736:
9719:
9717:
9716:
9711:
9706:
9705:
9696:
9695:
9679:
9677:
9676:
9671:
9666:
9665:
9654:
9653:
9643:
9642:
9637:
9636:
9619:
9617:
9616:
9611:
9609:
9608:
9593:
9592:
9587:
9586:
9567:
9566:
9555:
9554:
9534:
9532:
9531:
9526:
9518:
9517:
9508:
9487:
9485:
9484:
9481:{\displaystyle }
9479:
9455:
9453:
9452:
9447:
9442:
9441:
9436:
9435:
9416:
9415:
9404:
9403:
9383:
9381:
9380:
9375:
9373:
9368:
9367:
9362:
9361:
9354:
9346:
9345:
9344:
9339:
9338:
9329:
9303:
9301:
9300:
9295:
9293:
9292:
9291:
9290:
9272:
9270:
9269:
9264:
9262:
9261:
9260:
9255:
9254:
9245:
9228:
9226:
9225:
9220:
9218:
9217:
9216:
9215:
9191:
9189:
9188:
9183:
9150:
9148:
9147:
9142:
9124:
9122:
9121:
9116:
9095:
9093:
9092:
9087:
9082:
9081:
9056:
9054:
9053:
9048:
9036:
9034:
9033:
9028:
9026:
9025:
9009:
9007:
9006:
9001:
8999:
8998:
8993:
8992:
8974:
8972:
8971:
8966:
8964:
8963:
8950:
8948:
8947:
8942:
8940:
8939:
8926:
8924:
8923:
8918:
8913:
8912:
8903:
8902:
8886:
8884:
8883:
8878:
8876:
8875:
8863:
8862:
8849:
8847:
8846:
8841:
8839:
8838:
8837:
8832:
8831:
8825:
8808:
8806:
8805:
8800:
8798:
8797:
8779:
8777:
8776:
8771:
8769:
8768:
8752:
8750:
8749:
8744:
8742:
8741:
8736:
8735:
8709:
8707:
8706:
8701:
8693:
8688:
8687:
8681:
8668:
8666:
8665:
8660:
8658:
8657:
8642:
8641:
8623:
8622:
8601:
8599:
8598:
8593:
8591:
8590:
8589:
8584:
8583:
8577:
8560:
8558:
8557:
8552:
8550:
8549:
8536:
8534:
8533:
8528:
8526:
8525:
8509:
8507:
8506:
8501:
8499:
8498:
8483:
8481:
8480:
8475:
8473:
8472:
8459:
8457:
8456:
8451:
8449:
8448:
8435:
8433:
8432:
8427:
8425:
8424:
8411:
8409:
8408:
8403:
8401:
8400:
8384:
8382:
8381:
8376:
8374:
8373:
8360:
8358:
8357:
8352:
8350:
8349:
8336:
8334:
8333:
8328:
8316:
8314:
8313:
8308:
8296:
8294:
8293:
8288:
8268:
8267:
8255:
8254:
8194:
8192:
8191:
8186:
8171:
8169:
8168:
8163:
8161:
8160:
8117:
8115:
8114:
8109:
8107:
8106:
8105:
8104:
8080:
8078:
8077:
8072:
8057:
8055:
8054:
8049:
8041:
8040:
8031:
8030:
8029:
7998:
7996:
7995:
7990:
7976:
7974:
7973:
7968:
7966:
7965:
7950:over each step.
7949:
7947:
7946:
7941:
7929:
7927:
7926:
7921:
7919:
7918:
7905:
7903:
7902:
7897:
7895:
7894:
7893:
7877:, we can define
7876:
7874:
7873:
7870:{\displaystyle }
7868:
7844:
7842:
7841:
7836:
7834:
7833:
7821:and a partition
7820:
7818:
7817:
7812:
7793:
7791:
7790:
7785:
7783:
7782:
7781:
7780:
7756:
7754:
7753:
7748:
7733:
7731:
7730:
7725:
7723:
7722:
7717:
7708:
7697:
7693:
7692:
7687:
7686:
7671:
7666:
7661:
7660:
7645:
7637:
7636:
7624:
7623:
7607:
7602:
7545:
7543:
7542:
7537:
7513:
7511:
7510:
7505:
7490:
7488:
7487:
7482:
7477:
7476:
7458:
7457:
7442:
7441:
7428:
7426:
7425:
7420:
7378:
7376:
7375:
7370:
7324:
7322:
7321:
7316:
7311:
7307:
7306:
7301:
7300:
7291:
7285:
7280:
7260:
7256:
7252:
7251:
7239:
7238:
7222:
7171:
7169:
7168:
7163:
7161:
7156:
7155:
7146:
7138:
7133:
7132:
7123:
7096:
7094:
7093:
7088:
7076:
7074:
7073:
7068:
7056:
7054:
7053:
7048:
7009:
7007:
7006:
7001:
6989:
6987:
6986:
6981:
6976:
6975:
6963:
6962:
6943:
6941:
6940:
6935:
6918:
6917:
6899:
6898:
6882:
6880:
6879:
6874:
6862:
6860:
6859:
6854:
6842:
6840:
6839:
6834:
6816:
6814:
6813:
6808:
6776:be a graph with
6775:
6773:
6772:
6767:
6735:
6733:
6732:
6727:
6715:
6713:
6712:
6707:
6705:
6704:
6685:
6683:
6682:
6677:
6675:
6674:
6658:
6656:
6655:
6650:
6635:
6633:
6632:
6627:
6615:
6613:
6612:
6607:
6595:
6593:
6592:
6587:
6585:
6584:
6571:
6569:
6568:
6563:
6561:
6560:
6541:
6539:
6538:
6533:
6531:
6529:
6528:
6519:
6518:
6513:
6512:
6503:
6498:
6493:
6492:
6483:
6477:
6475:
6474:
6464:
6463:
6460:
6455:
6452:
6447:
6446:
6434:
6433:
6410:
6409:
6397:
6396:
6380:
6379:
6378:
6330:
6328:
6327:
6322:
6317:
6316:
6298:
6297:
6285:
6284:
6259:
6257:
6256:
6251:
6249:
6248:
6247:
6246:
6236:
6235:
6221:
6219:
6218:
6213:
6211:
6210:
6197:
6195:
6194:
6189:
6187:
6186:
6164:
6162:
6161:
6156:
6154:
6153:
6134:
6132:
6131:
6126:
6118:
6117:
6099:
6098:
6086:
6085:
6061:
6060:
6042:
6041:
6029:
6028:
6003:
6002:
5999:
5994:
5991:
5986:
5985:
5973:
5972:
5949:
5948:
5936:
5935:
5919:
5918:
5915:
5910:
5907:
5902:
5901:
5889:
5888:
5861:
5859:
5858:
5853:
5851:
5850:
5837:
5835:
5834:
5829:
5827:
5826:
5810:
5808:
5807:
5802:
5800:
5799:
5798:
5797:
5787:
5786:
5769:
5767:
5766:
5761:
5756:
5755:
5754:
5753:
5743:
5742:
5732:
5731:
5730:
5729:
5719:
5718:
5704:
5703:
5700:
5695:
5692:
5687:
5686:
5674:
5673:
5650:
5649:
5648:
5647:
5637:
5636:
5626:
5625:
5624:
5623:
5613:
5612:
5598:
5597:
5594:
5589:
5586:
5581:
5580:
5568:
5567:
5544:
5543:
5542:
5541:
5531:
5530:
5520:
5519:
5518:
5517:
5507:
5506:
5492:
5491:
5490:
5446:
5445:
5423:
5421:
5420:
5415:
5413:
5412:
5396:
5394:
5393:
5388:
5386:
5385:
5369:
5367:
5366:
5361:
5359:
5358:
5336:
5334:
5333:
5328:
5326:
5325:
5312:
5310:
5309:
5304:
5302:
5301:
5288:
5286:
5285:
5280:
5278:
5277:
5265:
5264:
5242:
5240:
5239:
5234:
5232:
5231:
5219:
5218:
5196:
5194:
5193:
5188:
5176:
5174:
5173:
5168:
5163:
5162:
5150:
5149:
5130:
5128:
5127:
5122:
5090:
5088:
5087:
5082:
5077:
5076:
5061:
5060:
5042:
5041:
5020:parts such that
5019:
5017:
5016:
5011:
5009:
5008:
4992:
4990:
4989:
4984:
4982:
4981:
4965:
4963:
4962:
4957:
4955:
4954:
4941:
4939:
4938:
4933:
4931:
4930:
4917:
4915:
4914:
4909:
4897:
4895:
4894:
4889:
4868:
4866:
4865:
4860:
4855:
4854:
4836:
4835:
4820:
4819:
4795:
4793:
4792:
4787:
4776:
4774:
4773:
4768:
4766:
4765:
4741:
4740:
4707:
4706:
4694:
4693:
4675:
4673:
4672:
4664:
4658:
4657:
4652:
4651:
4642:
4636:
4634:
4632:
4631:
4623:
4617:
4616:
4611:
4610:
4601:
4595:
4587:
4586:
4568:
4551:
4515:
4513:
4512:
4507:
4505:
4504:
4482:
4480:
4479:
4474:
4472:
4471:
4449:
4447:
4446:
4441:
4439:
4437:
4436:
4428:
4422:
4421:
4416:
4415:
4406:
4400:
4398:
4396:
4395:
4387:
4381:
4380:
4375:
4374:
4365:
4359:
4350:
4348:
4347:
4342:
4340:
4311:
4310:
4298:
4297:
4282:
4274:
4260:
4249:
4234:
4232:
4231:
4226:
4224:
4220:
4198:
4194:
4190:
4189:
4171:
4170:
4152:
4151:
4133:
4132:
4107:
4105:
4104:
4096:
4091:
4083:
4077:
4076:
4067:
4062:
4061:
4046:
4035:
4034:
4022:
3987:as above. Then,
3986:
3984:
3983:
3978:
3958:
3956:
3955:
3950:
3948:
3946:
3945:
3936:
3935:
3927:
3922:
3914:
3908:
3906:
3905:
3872:
3868:
3864:
3863:
3845:
3844:
3826:
3825:
3807:
3806:
3776:
3774:
3773:
3768:
3760:
3759:
3741:
3740:
3724:
3722:
3721:
3716:
3704:
3702:
3701:
3696:
3665:
3663:
3662:
3657:
3652:
3651:
3633:
3632:
3631:
3608:
3606:
3605:
3600:
3598:
3597:
3584:
3582:
3581:
3576:
3574:
3573:
3572:
3555:
3553:
3552:
3547:
3542:
3541:
3529:
3528:
3509:
3507:
3506:
3501:
3496:
3495:
3477:
3476:
3461:
3460:
3447:
3445:
3444:
3439:
3437:
3436:
3423:
3421:
3420:
3415:
3413:
3412:
3400:If each part of
3396:
3394:
3393:
3388:
3376:
3374:
3373:
3368:
3350:
3348:
3347:
3342:
3316:
3315:
3310:
3309:
3299:
3298:
3293:
3292:
3272:
3270:
3269:
3264:
3262:
3261:
3246:
3235:
3234:
3222:
3207:
3205:
3204:
3199:
3194:
3193:
3188:
3187:
3177:
3176:
3171:
3170:
3157:
3155:
3154:
3146:
3141:
3133:
3127:
3126:
3117:
3112:
3111:
3102:
3101:
3089:
3088:
3073:
3071:
3070:
3062:
3056:
3055:
3050:
3049:
3040:
3034:
3032:
3030:
3029:
3021:
3015:
3014:
3009:
3008:
2999:
2993:
2990:
2985:
2969:
2964:
2943:
2942:
2930:
2912:
2910:
2909:
2904:
2881:
2879:
2878:
2870:
2865:
2857:
2851:
2831:
2823:
2822:
2810:
2809:
2794:
2792:
2791:
2783:
2777:
2776:
2771:
2770:
2761:
2755:
2753:
2751:
2750:
2742:
2736:
2735:
2730:
2729:
2720:
2714:
2711:
2706:
2690:
2685:
2658:
2643:
2641:
2640:
2635:
2623:
2621:
2620:
2615:
2610:
2609:
2597:
2596:
2568:
2566:
2565:
2560:
2558:
2557:
2541:
2539:
2538:
2533:
2521:
2519:
2518:
2513:
2511:
2510:
2494:
2492:
2491:
2486:
2474:
2472:
2471:
2466:
2454:
2452:
2451:
2446:
2434:
2432:
2431:
2426:
2414:
2412:
2411:
2406:
2394:
2392:
2391:
2386:
2381:
2380:
2362:
2361:
2346:
2345:
2340:
2339:
2325:
2323:
2322:
2317:
2312:
2311:
2293:
2292:
2277:
2276:
2271:
2270:
2248:
2246:
2245:
2240:
2214:
2213:
2208:
2207:
2197:
2196:
2191:
2190:
2170:
2168:
2167:
2162:
2150:
2148:
2147:
2142:
2130:
2128:
2127:
2122:
2120:
2119:
2114:
2113:
2099:
2097:
2096:
2091:
2089:
2088:
2083:
2082:
2058:
2056:
2055:
2050:
2042:
2040:
2039:
2030:
2029:
2024:
2023:
2014:
2009:
2004:
2003:
1994:
1988:
1985:
1980:
1964:
1959:
1941:
1940:
1931:
1930:
1918:
1917:
1902:
1900:
1899:
1890:
1889:
1884:
1883:
1874:
1869:
1864:
1863:
1854:
1848:
1845:
1840:
1824:
1819:
1798:
1797:
1771:
1769:
1768:
1763:
1761:
1760:
1751:
1750:
1738:
1737:
1722:
1720:
1719:
1710:
1709:
1704:
1703:
1694:
1689:
1684:
1683:
1674:
1668:
1665:
1660:
1644:
1639:
1618:
1617:
1605:
1604:
1588:
1583:
1567:
1562:
1541:
1540:
1519:. Specifically,
1518:
1516:
1515:
1510:
1505:
1504:
1495:
1494:
1475:
1473:
1472:
1467:
1465:
1464:
1451:
1447:
1445:
1444:
1439:
1434:
1433:
1415:
1414:
1399:
1398:
1382:
1380:
1379:
1374:
1369:
1368:
1356:
1355:
1339:
1334:
1318:
1313:
1292:
1291:
1286:
1285:
1275:
1274:
1269:
1268:
1245:
1241:
1239:
1238:
1233:
1228:
1227:
1209:
1208:
1193:
1192:
1187:
1186:
1172:
1168:
1166:
1165:
1160:
1155:
1154:
1136:
1135:
1120:
1119:
1114:
1113:
1096:
1094:
1093:
1088:
1086:
1085:
1061:
1059:
1058:
1049:
1048:
1040:
1035:
1027:
1021:
988:
972:
970:
969:
964:
956:
948:
936:
930:
913:
911:
910:
905:
903:
902:
846:
840:
836:
826:
820:
816:
806:
800:
794:
790:
776:
770:
764:
758:
752:
740:
725:
723:
722:
717:
715:
714:
709:
691:
680:
675:
674:
665:
660:
655:
654:
645:
639:
638:
635:
630:
627:
622:
621:
609:
608:
579:
577:
576:
571:
558:
556:
555:
550:
545:
544:
526:
525:
510:
509:
496:
494:
493:
488:
476:
474:
473:
468:
449:
445:
437:
433:
425:
423:
422:
417:
406:
402:
347:
342:| ≥ ε|
335:
330:| ≥ ε|
323:
313:
301:
296:
290:
284:
273:
257:
251:
245:
227:
225:
224:
219:
217:
215:
214:
206:
201:
193:
187:
183:
159:
126:
110:
104:
87:
78:is a graph with
77:
71:
49:bipartite graphs
14902:
14901:
14897:
14896:
14895:
14893:
14892:
14891:
14872:
14871:
14859:
14841:
14778:
14776:Further reading
14773:
14772:
14719:
14715:
14700:
14696:
14642:
14638:
14590:
14586:
14558:
14554:
14520:SIAM J. Comput.
14513:
14509:
14494:
14490:
14466:
14462:
14448:
14426:
14422:
14390:
14386:
14335:
14331:
14280:
14276:
14214:
14210:
14166:
14162:
14130:10.1.1.378.8503
14112:
14101:
14097:
14057:
14051:
14047:
14009:
14005:
13983:
13979:
13939:
13935:
13921:
13896:
13892:
13886:
13864:
13860:
13834:
13830:
13791:
13787:
13752:
13748:
13713:
13704:
13656:
13652:
13628:
13624:
13573:
13566:
13511:
13507:
13474:
13470:
13427:
13423:
13364:
13357:
13352:
13328:totally bounded
13308:
13305:
13304:
13285:
13282:
13281:
13265:
13262:
13261:
13244:
13240:
13238:
13235:
13234:
13218:
13215:
13214:
13168:Endre Szemerédi
13123:
13093:
13079:
13065:
13062:
13061:
13044:
13040:
13028:
13024:
13015:
13011:
12993:
12989:
12980:
12976:
12968:
12965:
12964:
12947:
12942:
12941:
12933:
12927:
12923:
12921:
12918:
12917:
12898:
12890:
12887:
12886:
12867:
12849:
12844:
12843:
12835:
12823:
12818:
12817:
12808:
12804:
12795:
12791:
12773:
12769:
12760:
12756:
12745:
12733:
12721:
12718:
12717:
12701:
12700:
12694:
12689:
12688:
12679:
12675:
12666:
12662:
12644:
12640:
12631:
12627:
12616:
12604:
12595:
12586:
12581:
12580:
12572:
12568:
12563:
12554:
12553:
12547:
12542:
12541:
12532:
12528:
12519:
12515:
12497:
12493:
12484:
12480:
12469:
12464:
12455:
12450:
12449:
12441:
12437:
12432:
12420:
12407:
12406:
12400:
12395:
12394:
12385:
12381:
12372:
12368:
12350:
12346:
12337:
12333:
12322:
12317:
12309:
12305:
12299:
12293:
12289:
12284:
12279:
12273:
12269:
12264:
12263:
12261:
12249:
12236:
12235:
12226:
12225:
12207:
12206:
12193:
12186:
12184:
12181:
12180:
12159:
12151:
12148:
12147:
12130:
12126:
12124:
12121:
12120:
12103:
12099:
12097:
12094:
12093:
12073:
12068:
12067:
12059:
12055:
12050:
12045:
12042:
12041:
12024:
12020:
12018:
12015:
12014:
11997:
11993:
11991:
11988:
11987:
11970:
11969:
11967:
11964:
11963:
11943:
11938:
11937:
11929:
11925:
11919:
11913:
11909:
11904:
11899:
11893:
11889:
11884:
11883:
11881:
11875:
11871:
11861:
11848:
11847:
11846:
11837:
11832:
11831:
11823:
11822:
11817:
11815:
11812:
11811:
11794:
11793:
11791:
11788:
11787:
11766:
11761:
11753:
11750:
11749:
11727:
11724:
11723:
11706:
11702:
11700:
11697:
11696:
11679:
11678:
11676:
11673:
11672:
11655:
11651:
11649:
11646:
11645:
11628:
11624:
11622:
11619:
11618:
11601:
11597:
11595:
11592:
11591:
11574:
11570:
11568:
11565:
11564:
11544:
11540:
11525:
11521:
11510:
11507:
11506:
11481:
11475:
11474:
11469:
11467:
11464:
11463:
11437:
11436:
11418:
11417:
11409:
11406:
11405:
11388:
11387:
11385:
11382:
11381:
11364:
11359:
11358:
11350:
11345:
11340:
11337:
11336:
11319:
11318:
11316:
11313:
11312:
11296:
11293:
11292:
11275:
11274:
11272:
11269:
11268:
11249:
11243:
11238:
11226:
11223:
11222:
11185:
11182:
11181:
11163:
11157:
11156:
11151:
11150:
11146:
11144:
11141:
11140:
11120:
11116:
11107:
11103:
11098:
11095:
11094:
11091:
11082:
11043:
11040:
11039:
11022:
11017:
11016:
11010:
11009:
11004:
10998:
10994:
10992:
10989:
10988:
10971:
10967:
10959:
10950:
10946:
10937:
10933:
10915:
10911:
10902:
10898:
10887:
10885:
10882:
10881:
10846:
10843:
10842:
10824:
10818:
10817:
10812:
10811:
10807:
10805:
10802:
10801:
10781:
10777:
10768:
10764:
10759:
10756:
10755:
10729:
10723:
10719:
10714:
10712:
10709:
10708:
10689:
10686:
10685:
10668:
10664:
10655:
10651:
10649:
10646:
10645:
10627:
10623:
10602:
10598:
10597:
10588:
10587:
10585:
10582:
10581:
10559:
10556:
10555:
10554:, there exists
10520:
10516:
10507:
10503:
10501:
10498:
10497:
10494:
10489:
10467:
10466:
10464:
10461:
10460:
10443:
10442:
10440:
10437:
10436:
10420:
10417:
10416:
10413:
10384:
10376:
10368:
10360:
10358:
10355:
10354:
10337:
10333:
10319:
10315:
10310:
10300:
10288:
10285:
10284:
10262:
10259:
10258:
10241:
10237:
10227:
10221:
10217:
10212:
10211:
10207:
10186:
10182:
10180:
10177:
10176:
10159:
10155:
10145:
10139:
10135:
10130:
10129:
10125:
10111:
10105:
10099:
10098:
10097:
10092:
10082:
10076:
10072:
10067:
10066:
10062:
10048:
10036:
10032:
10027:
10025:
10022:
10021:
10004:
10000:
9992:
9986:
9982:
9977:
9975:
9972:
9971:
9954:
9950:
9942:
9936:
9930:
9929:
9928:
9923:
9921:
9918:
9917:
9897:
9893:
9878:
9874:
9872:
9869:
9868:
9851:
9847:
9839:
9833:
9827:
9826:
9825:
9820:
9818:
9815:
9814:
9797:
9793:
9791:
9788:
9787:
9786:, there exists
9768:
9765:
9764:
9738:
9732:
9731:
9730:
9728:
9725:
9724:
9701:
9700:
9691:
9690:
9685:
9682:
9681:
9655:
9649:
9648:
9647:
9638:
9632:
9631:
9630:
9625:
9622:
9621:
9604:
9600:
9588:
9582:
9581:
9580:
9556:
9550:
9549:
9548:
9540:
9537:
9536:
9513:
9509:
9504:
9493:
9490:
9489:
9461:
9458:
9457:
9437:
9431:
9430:
9429:
9405:
9399:
9398:
9397:
9389:
9386:
9385:
9369:
9363:
9357:
9356:
9355:
9350:
9340:
9334:
9330:
9325:
9324:
9320:
9309:
9306:
9305:
9286:
9282:
9281:
9280:
9278:
9275:
9274:
9256:
9250:
9246:
9241:
9240:
9236:
9234:
9231:
9230:
9205:
9201:
9200:
9199:
9197:
9194:
9193:
9159:
9156:
9155:
9130:
9127:
9126:
9101:
9098:
9097:
9077:
9073:
9062:
9059:
9058:
9042:
9039:
9038:
9021:
9017:
9015:
9012:
9011:
8994:
8988:
8987:
8986:
8984:
8981:
8980:
8959:
8958:
8956:
8953:
8952:
8935:
8934:
8932:
8929:
8928:
8908:
8907:
8898:
8897:
8892:
8889:
8888:
8871:
8867:
8858:
8857:
8855:
8852:
8851:
8833:
8827:
8826:
8821:
8820:
8816:
8814:
8811:
8810:
8793:
8792:
8790:
8787:
8786:
8764:
8760:
8758:
8755:
8754:
8737:
8731:
8730:
8729:
8727:
8724:
8723:
8716:
8689:
8683:
8682:
8677:
8675:
8672:
8671:
8653:
8649:
8637:
8636:
8618:
8617:
8609:
8606:
8605:
8585:
8579:
8578:
8573:
8572:
8568:
8566:
8563:
8562:
8545:
8544:
8542:
8539:
8538:
8521:
8517:
8515:
8512:
8511:
8494:
8493:
8491:
8488:
8487:
8468:
8467:
8465:
8462:
8461:
8444:
8443:
8441:
8438:
8437:
8420:
8419:
8417:
8414:
8413:
8396:
8395:
8393:
8390:
8389:
8369:
8368:
8366:
8363:
8362:
8345:
8344:
8342:
8339:
8338:
8322:
8319:
8318:
8302:
8299:
8298:
8263:
8259:
8250:
8246:
8244:
8241:
8240:
8237:
8213:
8180:
8177:
8176:
8156:
8152:
8147:
8144:
8143:
8124:
8097:
8093:
8092:
8088:
8086:
8083:
8082:
8066:
8063:
8062:
8036:
8032:
8025:
8024:
8020:
8009:
8006:
8005:
7984:
7981:
7980:
7961:
7960:
7958:
7955:
7954:
7935:
7932:
7931:
7914:
7913:
7911:
7908:
7907:
7889:
7888:
7884:
7882:
7879:
7878:
7850:
7847:
7846:
7829:
7828:
7826:
7823:
7822:
7806:
7803:
7802:
7773:
7769:
7768:
7764:
7762:
7759:
7758:
7742:
7739:
7738:
7718:
7713:
7712:
7704:
7688:
7682:
7678:
7667:
7662:
7656:
7652:
7641:
7632:
7628:
7619:
7615:
7603:
7586:
7560:
7556:
7554:
7551:
7550:
7519:
7516:
7515:
7499:
7496:
7495:
7472:
7468:
7453:
7449:
7437:
7436:
7434:
7431:
7430:
7396:
7393:
7392:
7385:
7364:
7361:
7360:
7302:
7296:
7292:
7287:
7281:
7270:
7265:
7261:
7247:
7243:
7234:
7230:
7191:
7186:
7182:
7180:
7177:
7176:
7157:
7151:
7147:
7142:
7134:
7128:
7124:
7119:
7102:
7099:
7098:
7082:
7079:
7078:
7062:
7059:
7058:
7015:
7012:
7011:
6995:
6992:
6991:
6971:
6967:
6958:
6954:
6949:
6946:
6945:
6913:
6909:
6894:
6890:
6888:
6885:
6884:
6868:
6865:
6864:
6848:
6845:
6844:
6822:
6819:
6818:
6781:
6778:
6777:
6761:
6758:
6757:
6747:
6742:
6721:
6718:
6717:
6697:
6693:
6691:
6688:
6687:
6670:
6666:
6664:
6661:
6660:
6644:
6641:
6640:
6621:
6618:
6617:
6601:
6598:
6597:
6580:
6579:
6577:
6574:
6573:
6556:
6552:
6550:
6547:
6546:
6524:
6520:
6514:
6508:
6504:
6499:
6494:
6488:
6484:
6479:
6478:
6476:
6470:
6466:
6459:
6453: not
6451:
6442:
6438:
6429:
6425:
6421:
6405:
6401:
6392:
6388:
6374:
6370:
6345:
6339:
6336:
6335:
6306:
6302:
6293:
6289:
6274:
6270:
6265:
6262:
6261:
6242:
6238:
6237:
6231:
6230:
6229:
6227:
6224:
6223:
6206:
6205:
6203:
6200:
6199:
6176:
6172:
6170:
6167:
6166:
6149:
6145:
6143:
6140:
6139:
6107:
6103:
6094:
6090:
6075:
6071:
6050:
6046:
6037:
6033:
6018:
6014:
5998:
5992: not
5990:
5981:
5977:
5968:
5964:
5960:
5944:
5940:
5931:
5927:
5914:
5906:
5897:
5893:
5884:
5880:
5876:
5870:
5867:
5866:
5846:
5845:
5843:
5840:
5839:
5822:
5818:
5816:
5813:
5812:
5793:
5789:
5788:
5782:
5781:
5780:
5778:
5775:
5774:
5749:
5745:
5744:
5738:
5737:
5736:
5725:
5721:
5720:
5714:
5713:
5712:
5699:
5693: not
5691:
5682:
5678:
5669:
5665:
5661:
5643:
5639:
5638:
5632:
5631:
5630:
5619:
5615:
5614:
5608:
5607:
5606:
5593:
5585:
5576:
5572:
5563:
5559:
5555:
5537:
5533:
5532:
5526:
5525:
5524:
5513:
5509:
5508:
5502:
5501:
5500:
5486:
5482:
5457:
5441:
5440:
5432:
5429:
5428:
5408:
5404:
5402:
5399:
5398:
5381:
5377:
5375:
5372:
5371:
5348:
5344:
5342:
5339:
5338:
5321:
5320:
5318:
5315:
5314:
5297:
5296:
5294:
5291:
5290:
5273:
5269:
5254:
5250:
5248:
5245:
5244:
5227:
5223:
5208:
5204:
5202:
5199:
5198:
5197:-regular, find
5182:
5179:
5178:
5158:
5154:
5145:
5141:
5136:
5133:
5132:
5104:
5101:
5100:
5072:
5068:
5056:
5055:
5037:
5036:
5028:
5025:
5024:
5004:
5000:
4998:
4995:
4994:
4977:
4973:
4971:
4968:
4967:
4950:
4949:
4947:
4944:
4943:
4926:
4925:
4923:
4920:
4919:
4903:
4900:
4899:
4874:
4871:
4870:
4850:
4846:
4831:
4827:
4815:
4814:
4812:
4809:
4808:
4781:
4778:
4777:
4761:
4757:
4736:
4732:
4702:
4698:
4689:
4685:
4668:
4660:
4659:
4653:
4647:
4643:
4638:
4637:
4635:
4627:
4619:
4618:
4612:
4606:
4602:
4597:
4596:
4594:
4582:
4578:
4564:
4547:
4524:
4521:
4520:
4500:
4496:
4488:
4485:
4484:
4467:
4463:
4455:
4452:
4451:
4432:
4424:
4423:
4417:
4411:
4407:
4402:
4401:
4399:
4391:
4383:
4382:
4376:
4370:
4366:
4361:
4360:
4358:
4356:
4353:
4352:
4336:
4306:
4302:
4293:
4289:
4278:
4270:
4256:
4245:
4243:
4240:
4239:
4185:
4181:
4166:
4162:
4147:
4143:
4128:
4124:
4120:
4116:
4112:
4108:
4100:
4092:
4087:
4079:
4078:
4072:
4068:
4066:
4057:
4053:
4042:
4030:
4026:
4018:
3995:
3992:
3991:
3972:
3969:
3968:
3941:
3937:
3931:
3923:
3918:
3910:
3909:
3907:
3901:
3897:
3859:
3855:
3840:
3836:
3821:
3817:
3802:
3798:
3794:
3790:
3785:
3782:
3781:
3755:
3751:
3736:
3732:
3730:
3727:
3726:
3710:
3707:
3706:
3678:
3675:
3674:
3647:
3646:
3624:
3623:
3622:
3614:
3611:
3610:
3593:
3592:
3590:
3587:
3586:
3565:
3564:
3563:
3561:
3558:
3557:
3537:
3533:
3524:
3520:
3515:
3512:
3511:
3491:
3487:
3472:
3468:
3456:
3455:
3453:
3450:
3449:
3432:
3431:
3429:
3426:
3425:
3408:
3407:
3405:
3402:
3401:
3382:
3379:
3378:
3356:
3353:
3352:
3311:
3305:
3304:
3303:
3294:
3288:
3287:
3286:
3278:
3275:
3274:
3257:
3253:
3242:
3230:
3226:
3218:
3216:
3213:
3212:
3189:
3183:
3182:
3181:
3172:
3166:
3165:
3164:
3150:
3142:
3137:
3129:
3128:
3122:
3118:
3116:
3107:
3103:
3097:
3093:
3084:
3080:
3066:
3058:
3057:
3051:
3045:
3041:
3036:
3035:
3033:
3025:
3017:
3016:
3010:
3004:
3000:
2995:
2994:
2992:
2986:
2975:
2965:
2954:
2938:
2934:
2926:
2924:
2921:
2920:
2874:
2866:
2861:
2853:
2852:
2832:
2830:
2818:
2814:
2805:
2801:
2787:
2779:
2778:
2772:
2766:
2762:
2757:
2756:
2754:
2746:
2738:
2737:
2731:
2725:
2721:
2716:
2715:
2713:
2707:
2696:
2686:
2675:
2654:
2652:
2649:
2648:
2629:
2626:
2625:
2605:
2601:
2592:
2588:
2574:
2571:
2570:
2553:
2549:
2547:
2544:
2543:
2527:
2524:
2523:
2506:
2502:
2500:
2497:
2496:
2480:
2477:
2476:
2460:
2457:
2456:
2455:uniformly from
2440:
2437:
2436:
2420:
2417:
2416:
2415:uniformly from
2400:
2397:
2396:
2376:
2372:
2357:
2353:
2341:
2335:
2334:
2333:
2331:
2328:
2327:
2307:
2303:
2288:
2284:
2272:
2266:
2265:
2264:
2262:
2259:
2258:
2209:
2203:
2202:
2201:
2192:
2186:
2185:
2184:
2176:
2173:
2172:
2156:
2153:
2152:
2136:
2133:
2132:
2131:of vertex sets
2115:
2109:
2108:
2107:
2105:
2102:
2101:
2084:
2078:
2077:
2076:
2074:
2071:
2070:
2035:
2031:
2025:
2019:
2015:
2010:
2005:
1999:
1995:
1990:
1989:
1987:
1981:
1970:
1960:
1949:
1936:
1932:
1926:
1922:
1913:
1909:
1895:
1891:
1885:
1879:
1875:
1870:
1865:
1859:
1855:
1850:
1849:
1847:
1841:
1830:
1820:
1809:
1793:
1792:
1784:
1781:
1780:
1756:
1752:
1746:
1742:
1733:
1729:
1715:
1711:
1705:
1699:
1695:
1690:
1685:
1679:
1675:
1670:
1669:
1667:
1661:
1650:
1640:
1629:
1613:
1609:
1600:
1596:
1584:
1573:
1563:
1552:
1536:
1535:
1527:
1524:
1523:
1500:
1499:
1490:
1489:
1481:
1478:
1477:
1460:
1459:
1457:
1454:
1453:
1449:
1429:
1425:
1410:
1406:
1394:
1393:
1391:
1388:
1387:
1364:
1360:
1351:
1347:
1335:
1324:
1314:
1303:
1287:
1281:
1280:
1279:
1270:
1264:
1263:
1262:
1254:
1251:
1250:
1243:
1223:
1219:
1204:
1200:
1188:
1182:
1181:
1180:
1178:
1175:
1174:
1170:
1150:
1146:
1131:
1127:
1115:
1109:
1108:
1107:
1105:
1102:
1101:
1100:For partitions
1081:
1077:
1054:
1050:
1044:
1036:
1031:
1023:
1022:
1020:
997:
994:
993:
989:is defined as:
978:
952:
944:
942:
939:
938:
932:
922:
898:
894:
892:
889:
888:
853:
842:
838:
832:
822:
818:
812:
802:
796:
792:
778:
772:
766:
760:
754:
748:
738:
710:
705:
704:
687:
676:
670:
666:
661:
656:
650:
646:
641:
634:
628: not
626:
617:
613:
604:
600:
596:
590:
587:
586:
565:
562:
561:
540:
536:
521:
517:
505:
504:
502:
499:
498:
482:
479:
478:
462:
459:
458:
457:A partition of
447:
443:
435:
431:
362:
358:
356:
353:
352:
337:
325:
315:
305:
299:
292:
286:
282:
271:
253:
247:
232:
210:
202:
197:
189:
188:
164:
160:
158:
135:
132:
131:
127:is defined as:
116:
106:
96:
83:
73:
69:
65:
45:Endre Szemerédi
17:
12:
11:
5:
14900:
14890:
14889:
14884:
14870:
14869:
14858:
14857:External links
14855:
14854:
14853:
14839:
14803:
14787:Simonovits, M.
14777:
14774:
14771:
14770:
14722:Lovász, László
14713:
14694:
14636:
14584:
14552:
14526:(2): 505–522,
14507:
14488:
14460:
14446:
14420:
14407:10.1.1.102.681
14384:
14356:(3): 297–312,
14329:
14303:(1): 109–123,
14274:
14229:(3): 897–946,
14208:
14160:
14123:(2): 113–179,
14095:
14045:
14026:(2): 131–164,
14003:
13977:
13933:
13919:
13899:Hajnal, András
13890:
13884:
13858:
13828:
13810:(2): 212–243,
13785:
13767:(4): 798–859,
13746:
13728:(2): 175–220,
13702:
13650:
13634:(MSc thesis),
13622:
13596:(4): 451–476,
13584:Szegedy, Mario
13564:
13505:
13487:(1): 104–109,
13468:
13442:(2): 322–337,
13421:
13354:
13353:
13351:
13348:
13312:
13289:
13269:
13247:
13243:
13222:
13142:Szemerédi 1978
13122:
13119:
13106:
13103:
13100:
13096:
13092:
13089:
13086:
13082:
13078:
13075:
13072:
13069:
13047:
13043:
13039:
13036:
13031:
13027:
13023:
13018:
13014:
13010:
13007:
13004:
13001:
12996:
12992:
12988:
12983:
12979:
12975:
12972:
12950:
12945:
12940:
12936:
12930:
12926:
12905:
12901:
12897:
12894:
12874:
12870:
12866:
12863:
12860:
12857:
12852:
12847:
12842:
12838:
12834:
12831:
12826:
12821:
12816:
12811:
12807:
12803:
12798:
12794:
12790:
12787:
12784:
12781:
12776:
12772:
12768:
12763:
12759:
12755:
12752:
12748:
12742:
12739:
12736:
12732:
12728:
12725:
12697:
12692:
12687:
12682:
12678:
12674:
12669:
12665:
12661:
12658:
12655:
12652:
12647:
12643:
12639:
12634:
12630:
12626:
12623:
12619:
12613:
12610:
12607:
12603:
12598:
12589:
12584:
12579:
12575:
12571:
12567:
12562:
12559:
12557:
12555:
12550:
12545:
12540:
12535:
12531:
12527:
12522:
12518:
12514:
12511:
12508:
12505:
12500:
12496:
12492:
12487:
12483:
12479:
12476:
12472:
12467:
12458:
12453:
12448:
12444:
12440:
12436:
12429:
12426:
12423:
12419:
12415:
12412:
12410:
12408:
12403:
12398:
12393:
12388:
12384:
12380:
12375:
12371:
12367:
12364:
12361:
12358:
12353:
12349:
12345:
12340:
12336:
12332:
12329:
12325:
12320:
12312:
12308:
12302:
12296:
12292:
12287:
12282:
12276:
12272:
12267:
12258:
12255:
12252:
12248:
12244:
12241:
12239:
12237:
12234:
12229:
12224:
12221:
12218:
12215:
12210:
12205:
12202:
12199:
12196:
12194:
12192:
12189:
12188:
12177:. Note that
12166:
12162:
12158:
12155:
12133:
12129:
12106:
12102:
12076:
12071:
12066:
12062:
12058:
12054:
12049:
12027:
12023:
12000:
11996:
11973:
11946:
11941:
11936:
11932:
11928:
11922:
11916:
11912:
11907:
11902:
11896:
11892:
11887:
11878:
11874:
11870:
11864:
11859:
11856:
11851:
11840:
11835:
11830:
11826:
11821:
11797:
11772:
11769:
11765:
11760:
11757:
11734:
11731:
11709:
11705:
11682:
11658:
11654:
11631:
11627:
11604:
11600:
11577:
11573:
11552:
11547:
11543:
11539:
11536:
11533:
11528:
11524:
11520:
11517:
11514:
11491:
11488:
11484:
11478:
11472:
11451:
11448:
11445:
11440:
11435:
11432:
11429:
11426:
11421:
11416:
11413:
11391:
11367:
11362:
11357:
11353:
11348:
11344:
11322:
11300:
11278:
11256:
11252:
11246:
11241:
11237:
11233:
11230:
11207:
11204:
11201:
11198:
11195:
11192:
11189:
11166:
11160:
11154:
11149:
11128:
11123:
11119:
11115:
11110:
11106:
11102:
11090:
11087:
11081:
11078:
11077:
11076:
11065:
11062:
11059:
11056:
11053:
11050:
11047:
11025:
11020:
11013:
11007:
11001:
10997:
10974:
10970:
10966:
10962:
10958:
10953:
10949:
10945:
10940:
10936:
10932:
10929:
10926:
10923:
10918:
10914:
10910:
10905:
10901:
10897:
10894:
10890:
10879:
10868:
10865:
10862:
10859:
10856:
10853:
10850:
10827:
10821:
10815:
10810:
10789:
10784:
10780:
10776:
10771:
10767:
10763:
10753:
10742:
10739:
10736:
10732:
10726:
10722:
10717:
10693:
10671:
10667:
10663:
10658:
10654:
10630:
10626:
10622:
10619:
10616:
10613:
10610:
10605:
10601:
10596:
10591:
10569:
10566:
10563:
10543:
10540:
10537:
10534:
10531:
10528:
10523:
10519:
10515:
10510:
10506:
10493:
10490:
10488:
10485:
10470:
10446:
10424:
10412:
10409:
10397:
10394:
10391:
10387:
10383:
10379:
10375:
10371:
10367:
10363:
10340:
10336:
10330:
10327:
10322:
10318:
10313:
10309:
10306:
10303:
10299:
10295:
10292:
10272:
10269:
10266:
10244:
10240:
10236:
10230:
10224:
10220:
10215:
10210:
10206:
10203:
10200:
10195:
10192:
10189:
10185:
10162:
10158:
10154:
10148:
10142:
10138:
10133:
10128:
10124:
10121:
10118:
10114:
10108:
10102:
10095:
10091:
10085:
10079:
10075:
10070:
10065:
10061:
10058:
10055:
10051:
10045:
10042:
10039:
10035:
10030:
10007:
10003:
9999:
9995:
9989:
9985:
9980:
9957:
9953:
9949:
9945:
9939:
9933:
9926:
9905:
9900:
9896:
9892:
9889:
9886:
9881:
9877:
9854:
9850:
9846:
9842:
9836:
9830:
9823:
9800:
9796:
9775:
9772:
9752:
9747:
9744:
9741:
9735:
9709:
9704:
9699:
9694:
9689:
9669:
9664:
9661:
9658:
9652:
9646:
9641:
9635:
9629:
9607:
9603:
9599:
9596:
9591:
9585:
9579:
9576:
9573:
9570:
9565:
9562:
9559:
9553:
9547:
9544:
9524:
9521:
9516:
9512:
9507:
9503:
9500:
9497:
9477:
9474:
9471:
9468:
9465:
9445:
9440:
9434:
9428:
9425:
9422:
9419:
9414:
9411:
9408:
9402:
9396:
9393:
9372:
9366:
9360:
9353:
9349:
9343:
9337:
9333:
9328:
9323:
9319:
9316:
9313:
9289:
9285:
9259:
9253:
9249:
9244:
9239:
9214:
9211:
9208:
9204:
9181:
9178:
9175:
9172:
9169:
9166:
9163:
9140:
9137:
9134:
9114:
9111:
9108:
9105:
9085:
9080:
9076:
9072:
9069:
9066:
9046:
9024:
9020:
8997:
8991:
8979:We start with
8977:
8976:
8962:
8938:
8916:
8911:
8906:
8901:
8896:
8874:
8870:
8866:
8861:
8836:
8830:
8824:
8819:
8796:
8782:
8781:
8767:
8763:
8740:
8734:
8715:
8712:
8711:
8710:
8699:
8696:
8692:
8686:
8680:
8669:
8656:
8652:
8648:
8645:
8640:
8635:
8632:
8629:
8626:
8621:
8616:
8613:
8603:
8588:
8582:
8576:
8571:
8548:
8524:
8520:
8497:
8485:
8471:
8447:
8423:
8399:
8372:
8348:
8326:
8306:
8286:
8283:
8280:
8277:
8274:
8271:
8266:
8262:
8258:
8253:
8249:
8236:
8233:
8212:
8209:
8184:
8159:
8155:
8151:
8123:
8120:
8103:
8100:
8096:
8091:
8070:
8059:
8058:
8047:
8044:
8039:
8035:
8028:
8023:
8019:
8016:
8013:
7988:
7964:
7939:
7917:
7892:
7887:
7866:
7863:
7860:
7857:
7854:
7832:
7810:
7779:
7776:
7772:
7767:
7746:
7735:
7734:
7721:
7716:
7711:
7707:
7703:
7700:
7696:
7691:
7685:
7681:
7677:
7674:
7670:
7665:
7659:
7655:
7651:
7648:
7644:
7640:
7635:
7631:
7627:
7622:
7618:
7614:
7611:
7606:
7601:
7598:
7595:
7592:
7589:
7585:
7581:
7578:
7575:
7572:
7569:
7566:
7563:
7559:
7535:
7532:
7529:
7526:
7523:
7503:
7491:is said to be
7480:
7475:
7471:
7467:
7464:
7461:
7456:
7452:
7448:
7445:
7440:
7418:
7415:
7412:
7409:
7406:
7403:
7400:
7391:Given a graph
7384:
7381:
7368:
7326:
7325:
7314:
7310:
7305:
7299:
7295:
7290:
7284:
7279:
7276:
7273:
7269:
7264:
7259:
7255:
7250:
7246:
7242:
7237:
7233:
7229:
7226:
7221:
7218:
7215:
7212:
7209:
7206:
7203:
7200:
7197:
7194:
7190:
7185:
7160:
7154:
7150:
7145:
7141:
7137:
7131:
7127:
7122:
7118:
7115:
7112:
7109:
7106:
7086:
7066:
7046:
7043:
7040:
7037:
7034:
7031:
7028:
7025:
7022:
7019:
6999:
6979:
6974:
6970:
6966:
6961:
6957:
6953:
6933:
6930:
6927:
6924:
6921:
6916:
6912:
6908:
6905:
6902:
6897:
6893:
6872:
6852:
6832:
6829:
6826:
6806:
6803:
6800:
6797:
6794:
6791:
6788:
6785:
6765:
6746:
6743:
6741:
6738:
6725:
6703:
6700:
6696:
6673:
6669:
6648:
6625:
6605:
6583:
6559:
6555:
6543:
6542:
6527:
6523:
6517:
6511:
6507:
6502:
6497:
6491:
6487:
6482:
6473:
6469:
6458:
6450:
6445:
6441:
6437:
6432:
6428:
6424:
6420:
6416:
6413:
6408:
6404:
6400:
6395:
6391:
6387:
6384:
6377:
6373:
6369:
6366:
6363:
6360:
6357:
6354:
6351:
6348:
6344:
6320:
6315:
6312:
6309:
6305:
6301:
6296:
6292:
6288:
6283:
6280:
6277:
6273:
6269:
6245:
6241:
6234:
6209:
6198:when creating
6185:
6182:
6179:
6175:
6152:
6148:
6136:
6135:
6124:
6121:
6116:
6113:
6110:
6106:
6102:
6097:
6093:
6089:
6084:
6081:
6078:
6074:
6070:
6067:
6064:
6059:
6056:
6053:
6049:
6045:
6040:
6036:
6032:
6027:
6024:
6021:
6017:
6013:
6010:
6007:
5997:
5989:
5984:
5980:
5976:
5971:
5967:
5963:
5959:
5955:
5952:
5947:
5943:
5939:
5934:
5930:
5926:
5923:
5913:
5905:
5900:
5896:
5892:
5887:
5883:
5879:
5875:
5849:
5825:
5821:
5796:
5792:
5785:
5771:
5770:
5759:
5752:
5748:
5741:
5735:
5728:
5724:
5717:
5711:
5708:
5698:
5690:
5685:
5681:
5677:
5672:
5668:
5664:
5660:
5656:
5653:
5646:
5642:
5635:
5629:
5622:
5618:
5611:
5605:
5602:
5592:
5584:
5579:
5575:
5571:
5566:
5562:
5558:
5554:
5550:
5547:
5540:
5536:
5529:
5523:
5516:
5512:
5505:
5499:
5496:
5489:
5485:
5481:
5478:
5475:
5472:
5469:
5466:
5463:
5460:
5456:
5452:
5449:
5444:
5439:
5436:
5411:
5407:
5384:
5380:
5357:
5354:
5351:
5347:
5324:
5300:
5276:
5272:
5268:
5263:
5260:
5257:
5253:
5230:
5226:
5222:
5217:
5214:
5211:
5207:
5186:
5166:
5161:
5157:
5153:
5148:
5144:
5140:
5120:
5117:
5114:
5111:
5108:
5092:
5091:
5080:
5075:
5071:
5067:
5064:
5059:
5054:
5051:
5048:
5045:
5040:
5035:
5032:
5007:
5003:
4980:
4976:
4953:
4929:
4907:
4887:
4884:
4881:
4878:
4858:
4853:
4849:
4845:
4842:
4839:
4834:
4830:
4826:
4823:
4818:
4797:
4796:
4785:
4764:
4760:
4756:
4753:
4750:
4747:
4744:
4739:
4735:
4731:
4728:
4725:
4722:
4719:
4716:
4713:
4710:
4705:
4701:
4697:
4692:
4688:
4684:
4681:
4678:
4671:
4667:
4663:
4656:
4650:
4646:
4641:
4630:
4626:
4622:
4615:
4609:
4605:
4600:
4593:
4590:
4585:
4581:
4577:
4574:
4571:
4567:
4563:
4560:
4557:
4554:
4550:
4546:
4543:
4540:
4537:
4534:
4531:
4528:
4503:
4499:
4495:
4492:
4470:
4466:
4462:
4459:
4435:
4431:
4427:
4420:
4414:
4410:
4405:
4394:
4390:
4386:
4379:
4373:
4369:
4364:
4339:
4335:
4332:
4329:
4326:
4323:
4320:
4317:
4314:
4309:
4305:
4301:
4296:
4292:
4288:
4285:
4281:
4277:
4273:
4269:
4266:
4263:
4259:
4255:
4252:
4248:
4236:
4235:
4223:
4219:
4216:
4213:
4210:
4207:
4204:
4201:
4197:
4193:
4188:
4184:
4180:
4177:
4174:
4169:
4165:
4161:
4158:
4155:
4150:
4146:
4142:
4139:
4136:
4131:
4127:
4123:
4119:
4115:
4111:
4103:
4099:
4095:
4090:
4086:
4082:
4075:
4071:
4065:
4060:
4056:
4052:
4049:
4045:
4041:
4038:
4033:
4029:
4025:
4021:
4017:
4014:
4011:
4008:
4005:
4002:
3999:
3976:
3960:
3959:
3944:
3940:
3934:
3930:
3926:
3921:
3917:
3913:
3904:
3900:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3871:
3867:
3862:
3858:
3854:
3851:
3848:
3843:
3839:
3835:
3832:
3829:
3824:
3820:
3816:
3813:
3810:
3805:
3801:
3797:
3793:
3789:
3766:
3763:
3758:
3754:
3750:
3747:
3744:
3739:
3735:
3714:
3694:
3691:
3688:
3685:
3682:
3655:
3650:
3645:
3642:
3639:
3636:
3630:
3627:
3621:
3618:
3596:
3571:
3568:
3545:
3540:
3536:
3532:
3527:
3523:
3519:
3499:
3494:
3490:
3486:
3483:
3480:
3475:
3471:
3467:
3464:
3459:
3435:
3411:
3386:
3366:
3363:
3360:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3314:
3308:
3302:
3297:
3291:
3285:
3282:
3260:
3256:
3252:
3249:
3245:
3241:
3238:
3233:
3229:
3225:
3221:
3211:By convexity,
3209:
3208:
3197:
3192:
3186:
3180:
3175:
3169:
3163:
3160:
3153:
3149:
3145:
3140:
3136:
3132:
3125:
3121:
3115:
3110:
3106:
3100:
3096:
3092:
3087:
3083:
3079:
3076:
3069:
3065:
3061:
3054:
3048:
3044:
3039:
3028:
3024:
3020:
3013:
3007:
3003:
2998:
2989:
2984:
2981:
2978:
2974:
2968:
2963:
2960:
2957:
2953:
2949:
2946:
2941:
2937:
2933:
2929:
2914:
2913:
2902:
2899:
2896:
2893:
2890:
2887:
2884:
2877:
2873:
2869:
2864:
2860:
2856:
2850:
2847:
2844:
2841:
2838:
2835:
2829:
2826:
2821:
2817:
2813:
2808:
2804:
2800:
2797:
2790:
2786:
2782:
2775:
2769:
2765:
2760:
2749:
2745:
2741:
2734:
2728:
2724:
2719:
2710:
2705:
2702:
2699:
2695:
2689:
2684:
2681:
2678:
2674:
2670:
2667:
2664:
2661:
2657:
2633:
2613:
2608:
2604:
2600:
2595:
2591:
2587:
2584:
2581:
2578:
2556:
2552:
2531:
2509:
2505:
2484:
2464:
2444:
2424:
2404:
2384:
2379:
2375:
2371:
2368:
2365:
2360:
2356:
2352:
2349:
2344:
2338:
2315:
2310:
2306:
2302:
2299:
2296:
2291:
2287:
2283:
2280:
2275:
2269:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2212:
2206:
2200:
2195:
2189:
2183:
2180:
2160:
2140:
2118:
2112:
2087:
2081:
2060:
2059:
2048:
2045:
2038:
2034:
2028:
2022:
2018:
2013:
2008:
2002:
1998:
1993:
1984:
1979:
1976:
1973:
1969:
1963:
1958:
1955:
1952:
1948:
1944:
1939:
1935:
1929:
1925:
1921:
1916:
1912:
1908:
1905:
1898:
1894:
1888:
1882:
1878:
1873:
1868:
1862:
1858:
1853:
1844:
1839:
1836:
1833:
1829:
1823:
1818:
1815:
1812:
1808:
1804:
1801:
1796:
1791:
1788:
1773:
1772:
1759:
1755:
1749:
1745:
1741:
1736:
1732:
1728:
1725:
1718:
1714:
1708:
1702:
1698:
1693:
1688:
1682:
1678:
1673:
1664:
1659:
1656:
1653:
1649:
1643:
1638:
1635:
1632:
1628:
1624:
1621:
1616:
1612:
1608:
1603:
1599:
1595:
1592:
1587:
1582:
1579:
1576:
1572:
1566:
1561:
1558:
1555:
1551:
1547:
1544:
1539:
1534:
1531:
1508:
1503:
1498:
1493:
1488:
1485:
1463:
1437:
1432:
1428:
1424:
1421:
1418:
1413:
1409:
1405:
1402:
1397:
1384:
1383:
1372:
1367:
1363:
1359:
1354:
1350:
1346:
1343:
1338:
1333:
1330:
1327:
1323:
1317:
1312:
1309:
1306:
1302:
1298:
1295:
1290:
1284:
1278:
1273:
1267:
1261:
1258:
1231:
1226:
1222:
1218:
1215:
1212:
1207:
1203:
1199:
1196:
1191:
1185:
1158:
1153:
1149:
1145:
1142:
1139:
1134:
1130:
1126:
1123:
1118:
1112:
1098:
1097:
1084:
1080:
1076:
1073:
1070:
1067:
1064:
1057:
1053:
1047:
1043:
1039:
1034:
1030:
1026:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
962:
959:
955:
951:
947:
931:be subsets of
901:
897:
880:
879:
876:
875:
874:
868:
852:
849:
727:
726:
713:
708:
703:
700:
697:
694:
690:
686:
683:
679:
673:
669:
664:
659:
653:
649:
644:
633:
625:
620:
616:
612:
607:
603:
599:
595:
569:
548:
543:
539:
535:
532:
529:
524:
520:
516:
513:
508:
486:
466:
427:
426:
415:
412:
409:
405:
401:
398:
395:
392:
389:
386:
383:
380:
377:
374:
371:
368:
365:
361:
229:
228:
213:
209:
205:
200:
196:
192:
186:
182:
179:
176:
173:
170:
167:
163:
157:
154:
151:
148:
145:
142:
139:
64:
61:
35:states that a
15:
9:
6:
4:
3:
2:
14899:
14888:
14885:
14883:
14880:
14879:
14877:
14868:
14865:
14861:
14860:
14850:
14846:
14842:
14836:
14832:
14828:
14824:
14820:
14816:
14812:
14808:
14804:
14800:
14796:
14792:
14788:
14784:
14780:
14779:
14767:
14763:
14759:
14755:
14751:
14747:
14742:
14737:
14733:
14729:
14728:
14723:
14717:
14710:
14709:
14704:
14698:
14690:
14686:
14681:
14676:
14672:
14668:
14663:
14658:
14654:
14650:
14646:
14640:
14633:
14629:
14625:
14621:
14617:
14613:
14608:
14603:
14599:
14595:
14588:
14581:
14577:
14572:
14567:
14563:
14556:
14549:
14545:
14541:
14537:
14533:
14529:
14525:
14521:
14517:
14511:
14504:
14503:
14498:
14492:
14484:
14483:10.37236/1449
14479:
14475:
14471:
14464:
14457:
14453:
14449:
14447:0-8186-7594-2
14443:
14439:
14435:
14431:
14424:
14417:
14413:
14408:
14403:
14399:
14395:
14388:
14381:
14377:
14373:
14369:
14364:
14359:
14355:
14351:
14347:
14343:
14339:
14338:Komlós, János
14333:
14326:
14322:
14318:
14314:
14310:
14306:
14302:
14298:
14297:
14296:Combinatorica
14292:
14288:
14284:
14283:Komlós, János
14278:
14270:
14266:
14262:
14258:
14254:
14250:
14246:
14242:
14237:
14232:
14228:
14224:
14223:
14218:
14217:Gowers, W. T.
14212:
14204:
14200:
14196:
14192:
14188:
14184:
14180:
14176:
14175:
14170:
14169:Gowers, W. T.
14164:
14156:
14152:
14148:
14144:
14140:
14136:
14131:
14126:
14122:
14118:
14110:
14106:
14105:Rödl, Vojtěch
14099:
14091:
14087:
14083:
14079:
14075:
14071:
14067:
14063:
14055:
14054:Rödl, Vojtěch
14049:
14041:
14037:
14033:
14029:
14025:
14021:
14017:
14016:Rödl, Vojtěch
14013:
14012:Frankl, Peter
14007:
13999:
13995:
13991:
13987:
13981:
13973:
13969:
13964:
13959:
13955:
13951:
13947:
13943:
13937:
13930:
13926:
13922:
13916:
13912:
13908:
13904:
13900:
13894:
13887:
13881:
13877:
13873:
13869:
13862:
13855:
13851:
13847:
13843:
13839:
13838:Rödl, Vojtěch
13832:
13825:
13821:
13817:
13813:
13809:
13805:
13804:
13799:
13795:
13789:
13782:
13778:
13774:
13770:
13766:
13762:
13761:
13756:
13755:Håstad, Johan
13750:
13743:
13739:
13735:
13731:
13727:
13723:
13722:
13721:Combinatorica
13717:
13711:
13709:
13707:
13699:
13695:
13691:
13687:
13683:
13679:
13674:
13669:
13665:
13661:
13654:
13646:
13641:
13637:
13633:
13626:
13619:
13615:
13611:
13607:
13603:
13599:
13595:
13591:
13590:
13589:Combinatorica
13585:
13581:
13577:
13571:
13569:
13561:
13557:
13553:
13549:
13544:
13539:
13534:
13529:
13525:
13521:
13520:
13515:
13509:
13502:
13498:
13494:
13490:
13486:
13482:
13478:
13472:
13465:
13461:
13457:
13453:
13449:
13445:
13441:
13437:
13436:
13431:
13430:Gowers, W. T.
13425:
13418:
13414:
13410:
13406:
13401:
13396:
13391:
13386:
13382:
13378:
13377:
13372:
13368:
13367:Conlon, David
13362:
13360:
13355:
13347:
13345:
13341:
13337:
13333:
13329:
13324:
13310:
13301:
13287:
13267:
13245:
13241:
13220:
13211:
13209:
13204:
13201:
13196:
13194:
13189:
13185:
13181:
13176:
13173:
13172:blow-up lemma
13169:
13165:
13164:Gábor Sárközy
13161:
13157:
13155:
13151:
13147:
13143:
13139:
13135:
13127:
13118:
13104:
13101:
13098:
13094:
13090:
13087:
13084:
13080:
13076:
13073:
13070:
13067:
13045:
13041:
13037:
13029:
13025:
13021:
13016:
13012:
13005:
13002:
12994:
12990:
12986:
12981:
12977:
12970:
12948:
12938:
12928:
12924:
12903:
12899:
12895:
12892:
12872:
12868:
12864:
12861:
12855:
12850:
12840:
12832:
12829:
12824:
12809:
12805:
12801:
12796:
12792:
12785:
12782:
12774:
12770:
12766:
12761:
12757:
12750:
12740:
12737:
12734:
12730:
12723:
12714:
12695:
12680:
12676:
12672:
12667:
12663:
12656:
12653:
12645:
12641:
12637:
12632:
12628:
12621:
12611:
12608:
12605:
12601:
12587:
12577:
12569:
12565:
12560:
12558:
12548:
12533:
12529:
12525:
12520:
12516:
12509:
12506:
12498:
12494:
12490:
12485:
12481:
12474:
12456:
12446:
12438:
12434:
12427:
12424:
12421:
12417:
12413:
12411:
12401:
12386:
12382:
12378:
12373:
12369:
12362:
12359:
12351:
12347:
12343:
12338:
12334:
12327:
12310:
12306:
12294:
12290:
12274:
12270:
12256:
12253:
12250:
12246:
12242:
12240:
12219:
12216:
12200:
12197:
12195:
12190:
12178:
12164:
12160:
12156:
12153:
12146:is irregular
12131:
12127:
12104:
12100:
12074:
12064:
12056:
12052:
12047:
12040:is irregular
12025:
12021:
11998:
11994:
11944:
11934:
11926:
11914:
11910:
11894:
11890:
11876:
11872:
11868:
11857:
11854:
11838:
11828:
11819:
11770:
11767:
11763:
11758:
11755:
11746:
11732:
11729:
11707:
11703:
11656:
11652:
11629:
11625:
11602:
11598:
11575:
11571:
11545:
11541:
11537:
11534:
11531:
11526:
11522:
11515:
11512:
11503:
11489:
11486:
11449:
11446:
11430:
11427:
11411:
11365:
11355:
11346:
11342:
11298:
11254:
11250:
11244:
11239:
11235:
11231:
11228:
11219:
11205:
11202:
11199:
11196:
11193:
11190:
11187:
11180:-regular for
11147:
11121:
11117:
11113:
11108:
11104:
11086:
11063:
11060:
11057:
11054:
11051:
11048:
11045:
11023:
10999:
10995:
10972:
10968:
10964:
10951:
10947:
10943:
10938:
10934:
10927:
10924:
10916:
10912:
10908:
10903:
10899:
10892:
10880:
10866:
10863:
10860:
10857:
10854:
10851:
10848:
10808:
10782:
10778:
10774:
10769:
10765:
10754:
10740:
10737:
10734:
10724:
10720:
10707:
10706:
10705:
10691:
10669:
10665:
10661:
10656:
10652:
10628:
10624:
10620:
10617:
10614:
10611:
10608:
10603:
10599:
10594:
10567:
10564:
10561:
10541:
10538:
10535:
10532:
10529:
10526:
10521:
10517:
10513:
10508:
10504:
10484:
10422:
10408:
10395:
10392:
10389:
10381:
10373:
10365:
10338:
10334:
10328:
10325:
10320:
10316:
10311:
10307:
10304:
10301:
10293:
10290:
10283:. By setting
10270:
10267:
10264:
10242:
10238:
10222:
10218:
10208:
10201:
10198:
10193:
10190:
10187:
10183:
10160:
10156:
10140:
10136:
10126:
10119:
10116:
10106:
10077:
10073:
10063:
10056:
10053:
10043:
10040:
10037:
10033:
10005:
10001:
9997:
9987:
9983:
9955:
9951:
9947:
9937:
9898:
9894:
9887:
9884:
9879:
9875:
9867:. By setting
9852:
9848:
9844:
9834:
9798:
9794:
9773:
9750:
9745:
9742:
9739:
9721:
9697:
9662:
9659:
9656:
9644:
9639:
9605:
9601:
9597:
9589:
9574:
9571:
9563:
9560:
9557:
9542:
9522:
9519:
9514:
9510:
9505:
9501:
9498:
9495:
9472:
9469:
9466:
9438:
9423:
9420:
9412:
9409:
9406:
9391:
9364:
9335:
9331:
9321:
9314:
9311:
9251:
9247:
9237:
9179:
9176:
9173:
9170:
9167:
9164:
9161:
9152:
9138:
9135:
9132:
9109:
9103:
9078:
9074:
9067:
9064:
9044:
9022:
9018:
8995:
8975:and continue.
8904:
8872:
8868:
8864:
8817:
8784:
8783:
8765:
8761:
8738:
8721:
8720:
8719:
8697:
8694:
8670:
8654:
8650:
8646:
8630:
8627:
8611:
8604:
8569:
8537:-regular and
8522:
8518:
8486:
8388:
8387:
8386:
8324:
8304:
8284:
8281:
8278:
8275:
8272:
8269:
8264:
8260:
8256:
8251:
8247:
8232:
8230:
8226:
8222:
8218:
8208:
8206:
8202:
8198:
8182:
8173:
8157:
8153:
8149:
8141:
8137:
8133:
8129:
8119:
8101:
8098:
8094:
8089:
8068:
8045:
8042:
8037:
8021:
8017:
8014:
8004:
8003:
8002:
8000:
7986:
7951:
7937:
7885:
7861:
7858:
7855:
7808:
7800:
7795:
7777:
7774:
7770:
7765:
7744:
7719:
7709:
7701:
7698:
7694:
7683:
7679:
7675:
7672:
7657:
7653:
7649:
7646:
7633:
7629:
7625:
7620:
7616:
7609:
7604:
7599:
7596:
7593:
7590:
7587:
7583:
7579:
7573:
7570:
7567:
7561:
7557:
7549:
7548:
7547:
7533:
7530:
7527:
7524:
7521:
7501:
7494:
7493:Frieze-Kannan
7473:
7469:
7465:
7462:
7459:
7454:
7450:
7443:
7413:
7410:
7407:
7401:
7398:
7389:
7380:
7366:
7358:
7357:sparse graphs
7353:
7351:
7346:
7344:
7340:
7336:
7332:
7312:
7308:
7297:
7293:
7282:
7277:
7274:
7271:
7267:
7262:
7257:
7248:
7244:
7240:
7235:
7231:
7224:
7216:
7210:
7207:
7201:
7198:
7195:
7188:
7183:
7175:
7174:
7173:
7152:
7148:
7139:
7129:
7125:
7116:
7110:
7104:
7084:
7064:
7041:
7035:
7032:
7026:
7023:
7020:
6997:
6972:
6968:
6964:
6959:
6955:
6928:
6922:
6919:
6914:
6910:
6906:
6903:
6900:
6895:
6891:
6870:
6850:
6830:
6827:
6824:
6801:
6795:
6789:
6783:
6763:
6755:
6750:
6737:
6723:
6701:
6698:
6694:
6671:
6667:
6646:
6636:
6623:
6603:
6557:
6553:
6525:
6521:
6509:
6505:
6489:
6485:
6471:
6467:
6456:
6443:
6439:
6435:
6430:
6426:
6418:
6414:
6406:
6402:
6398:
6393:
6389:
6382:
6375:
6367:
6361:
6355:
6352:
6349:
6342:
6334:
6333:
6332:
6313:
6310:
6307:
6303:
6294:
6290:
6286:
6281:
6278:
6275:
6271:
6243:
6239:
6183:
6180:
6177:
6173:
6150:
6146:
6114:
6111:
6108:
6104:
6095:
6091:
6087:
6082:
6079:
6076:
6072:
6065:
6057:
6054:
6051:
6047:
6038:
6034:
6030:
6025:
6022:
6019:
6015:
6005:
5995:
5982:
5978:
5974:
5969:
5965:
5957:
5953:
5945:
5941:
5937:
5932:
5928:
5921:
5911:
5898:
5894:
5890:
5885:
5881:
5873:
5865:
5864:
5863:
5823:
5819:
5794:
5790:
5750:
5746:
5733:
5726:
5722:
5706:
5696:
5683:
5679:
5675:
5670:
5666:
5658:
5654:
5644:
5640:
5627:
5620:
5616:
5600:
5590:
5577:
5573:
5569:
5564:
5560:
5552:
5548:
5538:
5534:
5521:
5514:
5510:
5494:
5487:
5479:
5473:
5467:
5464:
5461:
5454:
5450:
5434:
5427:
5426:
5425:
5409:
5405:
5382:
5378:
5355:
5352:
5349:
5345:
5274:
5270:
5266:
5261:
5258:
5255:
5251:
5228:
5224:
5220:
5215:
5212:
5209:
5205:
5184:
5159:
5155:
5151:
5146:
5142:
5115:
5112:
5109:
5098:
5078:
5073:
5069:
5065:
5049:
5046:
5030:
5023:
5022:
5021:
5005:
5001:
4978:
4974:
4905:
4882:
4876:
4851:
4847:
4843:
4840:
4837:
4832:
4828:
4821:
4806:
4801:
4783:
4762:
4758:
4754:
4751:
4748:
4745:
4742:
4737:
4726:
4723:
4720:
4714:
4711:
4703:
4699:
4695:
4690:
4686:
4679:
4665:
4648:
4644:
4624:
4607:
4603:
4591:
4583:
4572:
4561:
4558:
4544:
4538:
4532:
4529:
4526:
4519:
4518:
4517:
4501:
4497:
4493:
4490:
4468:
4464:
4460:
4457:
4429:
4412:
4408:
4388:
4371:
4367:
4330:
4327:
4324:
4318:
4315:
4307:
4303:
4299:
4294:
4290:
4283:
4275:
4264:
4253:
4250:
4221:
4214:
4211:
4208:
4202:
4199:
4195:
4186:
4182:
4175:
4172:
4167:
4163:
4156:
4148:
4144:
4137:
4134:
4129:
4125:
4117:
4113:
4109:
4097:
4084:
4073:
4069:
4063:
4058:
4050:
4039:
4031:
4027:
4015:
4009:
4003:
4000:
3997:
3990:
3989:
3988:
3974:
3966:
3942:
3938:
3928:
3915:
3902:
3898:
3894:
3888:
3885:
3882:
3876:
3873:
3869:
3860:
3856:
3849:
3846:
3841:
3837:
3830:
3822:
3818:
3811:
3808:
3803:
3799:
3791:
3787:
3780:
3779:
3778:
3764:
3761:
3756:
3752:
3748:
3745:
3742:
3737:
3733:
3712:
3689:
3686:
3683:
3672:
3667:
3640:
3637:
3628:
3616:
3569:
3538:
3534:
3530:
3525:
3521:
3492:
3488:
3484:
3481:
3478:
3473:
3469:
3462:
3397:
3384:
3364:
3361:
3358:
3335:
3332:
3329:
3323:
3320:
3312:
3300:
3295:
3280:
3258:
3250:
3239:
3231:
3227:
3190:
3178:
3173:
3158:
3147:
3134:
3123:
3119:
3113:
3108:
3098:
3094:
3090:
3085:
3081:
3074:
3063:
3046:
3042:
3022:
3005:
3001:
2987:
2982:
2979:
2976:
2972:
2966:
2961:
2958:
2955:
2951:
2947:
2939:
2935:
2919:
2918:
2917:
2897:
2894:
2891:
2885:
2882:
2871:
2858:
2845:
2842:
2839:
2833:
2827:
2819:
2815:
2811:
2806:
2802:
2795:
2784:
2767:
2763:
2743:
2726:
2722:
2708:
2703:
2700:
2697:
2693:
2687:
2682:
2679:
2676:
2672:
2668:
2662:
2647:
2646:
2645:
2631:
2606:
2602:
2598:
2593:
2589:
2582:
2579:
2576:
2554:
2550:
2529:
2507:
2503:
2482:
2462:
2442:
2422:
2402:
2377:
2373:
2369:
2366:
2363:
2358:
2354:
2347:
2342:
2308:
2304:
2300:
2297:
2294:
2289:
2285:
2278:
2273:
2256:
2250:
2233:
2230:
2227:
2221:
2218:
2210:
2198:
2193:
2178:
2158:
2138:
2116:
2085:
2068:
2063:
2046:
2043:
2036:
2032:
2020:
2016:
2000:
1996:
1982:
1977:
1974:
1971:
1967:
1961:
1956:
1953:
1950:
1946:
1942:
1937:
1927:
1923:
1919:
1914:
1910:
1903:
1896:
1892:
1880:
1876:
1860:
1856:
1842:
1837:
1834:
1831:
1827:
1821:
1816:
1813:
1810:
1806:
1802:
1786:
1779:
1778:
1777:
1757:
1747:
1743:
1739:
1734:
1730:
1723:
1716:
1712:
1700:
1696:
1680:
1676:
1662:
1657:
1654:
1651:
1647:
1641:
1636:
1633:
1630:
1626:
1622:
1614:
1610:
1606:
1601:
1597:
1590:
1585:
1580:
1577:
1574:
1570:
1564:
1559:
1556:
1553:
1549:
1545:
1529:
1522:
1521:
1520:
1496:
1483:
1430:
1426:
1422:
1419:
1416:
1411:
1407:
1400:
1365:
1361:
1357:
1352:
1348:
1341:
1336:
1331:
1328:
1325:
1321:
1315:
1310:
1307:
1304:
1300:
1296:
1288:
1276:
1271:
1256:
1249:
1248:
1247:
1224:
1220:
1216:
1213:
1210:
1205:
1201:
1194:
1189:
1151:
1147:
1143:
1140:
1137:
1132:
1128:
1121:
1116:
1082:
1074:
1071:
1068:
1062:
1055:
1051:
1041:
1028:
1017:
1011:
1008:
1005:
999:
992:
991:
990:
986:
982:
976:
960:
957:
949:
935:
929:
925:
920:
919:Definition 4.
915:
899:
895:
885:
877:
872:
871:
869:
866:
865:
864:
857:
848:
845:
835:
830:
829:Gowers (1997)
825:
815:
808:
805:
799:
789:
785:
781:
777:in the range
775:
769:
763:
759:such that if
757:
751:
747:
744:
736:
731:
711:
698:
692:
684:
681:
671:
667:
651:
647:
631:
618:
614:
610:
605:
601:
593:
585:
584:
583:
581:
567:
559:is called an
541:
537:
533:
530:
527:
522:
518:
511:
484:
464:
456:
455:Definition 3.
451:
441:
413:
410:
407:
403:
396:
393:
390:
384:
381:
375:
372:
369:
363:
359:
351:
350:
349:
345:
341:
333:
329:
322:
319: ⊆
318:
312:
309: ⊆
308:
303:
295:
289:
280:
279:Definition 2.
275:
264:
259:
256:
250:
243:
239:
235:
207:
194:
184:
177:
174:
171:
165:
161:
155:
149:
146:
143:
137:
130:
129:
128:
124:
120:
114:
109:
103:
99:
94:
93:Definition 1.
89:
86:
81:
76:
60:
58:
54:
50:
46:
42:
41:random graphs
38:
34:
30:
21:
14818:
14790:
14731:
14725:
14716:
14707:
14703:Tao, Terence
14697:
14662:math/0504472
14652:
14648:
14645:Tao, Terence
14639:
14597:
14593:
14587:
14571:math/0612838
14561:
14555:
14523:
14519:
14510:
14501:
14497:Tao, Terence
14491:
14473:
14469:
14463:
14429:
14423:
14397:
14393:
14387:
14363:math/9612213
14353:
14349:
14332:
14300:
14294:
14277:
14226:
14220:
14211:
14178:
14172:
14163:
14120:
14116:
14098:
14065:
14061:
14048:
14023:
14019:
14006:
13989:
13980:
13953:
13949:
13936:
13902:
13893:
13867:
13861:
13848:(1): 15–29,
13845:
13841:
13831:
13807:
13801:
13788:
13764:
13758:
13749:
13725:
13719:
13716:Frieze, Alan
13663:
13659:
13653:
13631:
13625:
13593:
13587:
13533:math/0503572
13523:
13522:, Series A,
13517:
13514:Tao, Terence
13508:
13484:
13480:
13471:
13439:
13433:
13424:
13380:
13374:
13336:cut distance
13325:
13302:
13212:
13205:
13197:
13177:
13160:János Komlós
13158:
13132:
12715:
12179:
11747:
11504:
11404:, such that
11220:
11092:
11083:
10987:for all but
10644:and subsets
10495:
10414:
9722:
9620:. We return
9153:
9096:parts. Here
8978:
8717:
8238:
8214:
8174:
8125:
8060:
7978:
7953:A partition
7952:
7796:
7736:
7492:
7390:
7386:
7354:
7347:
7328:
6753:
6752:
6748:
6740:Applications
6638:
6544:
6137:
5772:
5096:
5095:
4966:where every
4804:
4803:
4799:
4237:
3964:
3963:
3670:
3669:
3399:
3210:
2915:
2254:
2253:
2066:
2065:
2061:
1775:
1385:
1099:
984:
980:
977:of the pair
974:
933:
927:
923:
918:
917:
883:
881:
862:
843:
833:
823:
813:
810:
803:
797:
787:
783:
779:
773:
767:
761:
755:
749:
734:
733:
729:
560:
454:
453:
429:
343:
339:
331:
327:
320:
316:
310:
306:
298:
293:
287:
278:
277:
269:
254:
248:
241:
237:
233:
230:
122:
118:
115:of the pair
113:edge density
112:
107:
101:
97:
92:
91:
84:
74:
66:
32:
26:
14734:: 252–270,
14655:(1): 8–28,
14068:(1): 1–42,
13956:: 199–245,
13477:Roth, K. F.
13330:(and hence
13208:half graphs
13200:Terence Tao
13193:Terence Tao
13146:hypergraphs
11748:By setting
8722:Start with
8221:Krivelevich
8219:, Fischer,
8136:dense graph
8132:maximum cut
440:half graphs
324:satisfying
252:and one in
57:hypergraphs
14876:Categories
14807:Komlós, J.
14783:Komlós, J.
14600:: 80–145,
14516:Alon, Noga
14400:: 80–109,
13920:0897912640
13794:Alon, Noga
13673:1905.06917
13666:: 107070,
13645:1810.07275
13576:Alon, Noga
13371:Fox, Jacob
13350:References
13332:precompact
13140:, and in (
12916:, at most
11590:from each
11335:that is a
11291:that is a
11080:Motivation
10353:, we have
9916:, we have
9813:such that
9535:such that
8199:, such as
7097:is within
6944:such that
6817:, and let
6165:is cut by
5131:such that
3448:. Now, if
914:quantity.
811:The bound
737:For every
450:-regular.
348:, we have
297:is called
14750:1016-443X
14607:0801.1698
14540:0097-5397
14402:CiteSeerX
14236:0710.3032
14125:CiteSeerX
13698:155100313
13464:115242956
13390:1107.4829
13311:ε
13268:ε
13221:ε
13088:−
13074:−
13068:≥
13042:ϵ
13038:≥
13003:−
12925:ϵ
12893:≥
12862:≤
12830:≥
12783:−
12731:∑
12654:−
12602:∑
12507:−
12418:∑
12414:≥
12360:−
12247:∑
12217:−
12198:≥
12154:≤
12048:≤
11873:ϵ
11869:≤
11756:δ
11535:⋯
11487:≤
11447:≤
11428:−
11236:ϵ
11203:≤
11191:≤
11148:ϵ
11061:≤
11055:≤
11049:≤
10996:ϵ
10969:ϵ
10965:≤
10925:−
10864:≤
10858:≤
10852:≤
10809:ϵ
10738:δ
10684:for each
10662:⊂
10562:δ
10527:≥
10518:ϵ
10514:≥
10505:ϵ
10492:Statement
10390:≤
10317:ϵ
10305:≤
10209:ϵ
10127:ϵ
10117:≤
10064:ϵ
10054:≤
9998:≤
9948:≤
9895:ϵ
9845:≤
9771:∀
9602:ϵ
9572:−
9511:ϵ
9499:≤
9421:≥
9322:ϵ
9312:≤
9238:ϵ
9229:to be an
9192:, we set
9180:⋯
9133:ϵ
9075:ϵ
9065:≤
9019:ϵ
8869:ϵ
8865:≤
8818:ϵ
8762:ϵ
8695:≤
8651:ϵ
8602:-regular.
8570:ϵ
8519:ϵ
8270:≥
8261:ϵ
8257:≥
8248:ϵ
8235:Statement
8183:ϵ
8150:ϵ
8128:algorithm
8099:−
8095:ϵ
8069:ϵ
8046:ϵ
8043:≤
8038:◻
8034:‖
8018:−
8012:‖
7987:ϵ
7775:−
7771:ϵ
7745:ϵ
7702:ϵ
7699:≤
7676:∩
7650:∩
7584:∑
7580:−
7531:⊆
7502:ϵ
7463:…
7367:ε
7268:∏
7208:∈
7189:∏
7140:⋯
7117:ε
7033:∈
6998:ε
6920:⊆
6904:…
6825:ε
6724:ε
6699:−
6695:ε
6668:ε
6647:ε
6624:◻
6604:ε
6554:ε
6468:ε
6457:ε
6419:∑
6362:∈
6343:∑
6300:∖
6101:∖
6044:∖
5996:ε
5958:∑
5912:ε
5874:∑
5838:given by
5697:ε
5659:∑
5591:ε
5553:∑
5474:∈
5455:∑
5370:'s. Each
5267:⊂
5221:⊂
5185:ε
5070:ε
5047:≥
4906:ε
4841:…
4784:◻
4759:ε
4755:⋅
4752:ε
4749:⋅
4746:ε
4712:−
4592:≥
4562:−
4494:∈
4461:∈
4316:−
4254:−
4200:−
4179:∖
4141:∖
4040:−
3899:ε
3853:∖
3815:∖
3762:⊂
3743:⊂
3713:ε
3638:≥
3482:…
3385:◻
3321:≥
3240:≥
2973:∑
2952:∑
2694:∑
2673:∑
2367:…
2298:…
2219:≥
1968:∑
1947:∑
1943:≤
1828:∑
1807:∑
1648:∑
1627:∑
1571:∑
1550:∑
1420:…
1322:∑
1301:∑
1214:…
1141:…
685:ε
682:≤
632:ε
594:∑
568:ε
531:…
411:ε
408:≤
382:−
63:Statement
14766:15201345
14705:(2012),
14632:15421306
14499:(2009),
14456:38681854
14269:56118006
14203:14632612
14155:14126774
13944:(1975),
13929:17495443
13824:34786604
13742:15231198
13618:44645628
13560:14337591
13340:graphons
11617:and let
9154:Now for
8809:that is
8412:refines
7999:-regular
7799:graphons
6461:-regular
6000:-regular
5916:-regular
5701:-regular
5595:-regular
5099:For all
3777:, then,
3671:Lemma 2.
3629:′
3570:′
3351:for all
2542:in part
2495:in part
2067:Lemma 1.
743:positive
636:-regular
302:-regular
51:for his
14849:1966181
14799:1395865
14758:2306658
14689:2212136
14667:Bibcode
14612:Bibcode
14576:Bibcode
14548:2411033
14380:1635264
14325:6720143
14317:1466579
14261:2373376
14241:Bibcode
14195:2195580
14147:2198495
14090:7458739
14082:2069663
14040:1884430
13998:0540024
13972:0369312
13781:5120748
13678:Bibcode
13610:1804820
13552:2259060
13501:0051853
13456:1445389
13417:1623986
13409:2989432
13344:compact
9010:be an
8225:Szegedy
8140:NP-hard
7794:parts.
6596:is not
5177:is not
4898:is not
4805:Lemma 3
3967:Define
3705:is not
2475:, with
983:,
926:,
791:and an
746:integer
240:,
121:,
100:,
14847:
14837:
14797:
14764:
14756:
14748:
14687:
14630:
14546:
14538:
14454:
14444:
14404:
14378:
14323:
14315:
14267:
14259:
14201:
14193:
14153:
14145:
14127:
14088:
14080:
14038:
13996:
13970:
13927:
13917:
13882:
13822:
13779:
13740:
13696:
13616:
13608:
13558:
13550:
13499:
13462:
13454:
13415:
13407:
13188:Kannan
13184:Frieze
13154:Gowers
11139:to be
11038:pairs
8753:be an
8223:, and
6863:be an
6843:. Let
6572:since
6138:Since
5908:
5773:Where
5587:
5097:Proof:
4516:), so
3965:Proof:
2255:Proof:
1476:to be
975:energy
973:. The
937:. Set
231:where
111:. The
80:vertex
14762:S2CID
14657:arXiv
14628:S2CID
14602:arXiv
14566:arXiv
14452:S2CID
14358:arXiv
14321:S2CID
14265:S2CID
14231:arXiv
14199:S2CID
14151:S2CID
14086:S2CID
13925:S2CID
13820:S2CID
13777:S2CID
13738:S2CID
13694:S2CID
13668:arXiv
13640:arXiv
13614:S2CID
13556:S2CID
13528:arXiv
13460:S2CID
13413:S2CID
13385:arXiv
9304:with
9057:with
8951:with
8714:Proof
8134:in a
7979:weak
6222:, so
851:Proof
807:sets.
801:into
497:sets
477:into
37:graph
14835:ISBN
14746:ISSN
14536:ISSN
14442:ISBN
13915:ISBN
13880:ISBN
13186:and
13180:Rödl
13166:and
13150:Rödl
13102:>
12013:and
11730:>
11462:and
11221:Let
11197:<
10735:>
10565:>
10539:>
10459:and
9598:<
8628:<
8361:and
8282:>
8217:Alon
8203:and
8001:if:
6828:>
6756:Let
5243:and
4743:>
4483:and
3874:>
2522:and
2435:and
2326:and
2257:Let
2151:and
2100:and
1173:and
921:Let
819:O(ε)
741:and
291:and
281:For
95:Let
82:set
14827:doi
14736:doi
14675:doi
14620:doi
14528:doi
14478:doi
14434:doi
14412:doi
14368:doi
14305:doi
14249:doi
14227:166
14183:doi
14135:doi
14070:doi
14028:doi
13958:doi
13907:doi
13872:doi
13850:doi
13812:doi
13769:doi
13730:doi
13686:doi
13598:doi
13538:doi
13524:113
13489:doi
13444:doi
13395:doi
13117:.
11745:.
11671:in
11502:.
10800:is
10298:max
9720:.
9680:as
9151:.
8561:is
8510:is
8231:.
7977:is
7845:of
7172:of
7077:in
6990:is
5337:by
4942:of
4869:of
3585:of
1448:of
1242:of
1169:of
582:if
27:In
14878::
14845:MR
14843:,
14833:,
14821:,
14813:;
14795:MR
14785:;
14760:,
14754:MR
14752:,
14744:,
14732:17
14730:,
14685:MR
14683:,
14673:,
14665:,
14651:,
14626:,
14618:,
14610:,
14596:,
14574:,
14564:,
14544:MR
14542:,
14534:,
14524:38
14522:,
14472:,
14450:,
14440:,
14410:,
14398:16
14396:,
14376:MR
14374:,
14366:,
14354:12
14352:,
14344:;
14340:;
14319:,
14313:MR
14311:,
14301:17
14299:,
14289:;
14285:;
14263:,
14257:MR
14255:,
14247:,
14239:,
14197:,
14191:MR
14189:,
14179:15
14177:,
14149:,
14143:MR
14141:,
14133:,
14121:28
14119:,
14107:;
14084:,
14078:MR
14076:,
14066:25
14064:,
14036:MR
14034:,
14024:20
14022:,
14014:;
13994:MR
13968:MR
13966:,
13954:27
13952:,
13948:,
13923:,
13913:,
13878:,
13846:26
13844:,
13818:,
13808:67
13806:,
13775:,
13765:48
13763:,
13736:,
13726:19
13724:,
13705:^
13692:,
13684:,
13676:,
13664:98
13662:,
13638:,
13612:,
13606:MR
13604:,
13594:20
13592:,
13582:;
13567:^
13554:,
13548:MR
13546:,
13536:,
13497:MR
13495:,
13485:28
13483:,
13458:,
13452:MR
13450:,
13438:,
13411:,
13405:MR
13403:,
13393:,
13381:22
13379:,
13369:;
13358:^
13346:.
13300:.
13162:,
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2744:U
2740:|
2733:|
2727:i
2723:U
2718:|
2709:l
2704:1
2701:=
2698:j
2688:k
2683:1
2680:=
2677:i
2669:=
2666:]
2663:Z
2660:[
2656:E
2632:Z
2612:)
2607:j
2603:W
2599:,
2594:i
2590:U
2586:(
2583:d
2580:=
2577:Z
2555:j
2551:W
2530:y
2508:i
2504:U
2483:x
2463:W
2443:y
2423:U
2403:x
2383:}
2378:l
2374:W
2370:,
2364:,
2359:1
2355:W
2351:{
2348:=
2343:W
2337:P
2314:}
2309:k
2305:U
2301:,
2295:,
2290:1
2286:U
2282:{
2279:=
2274:U
2268:P
2237:)
2234:W
2231:,
2228:U
2225:(
2222:q
2216:)
2211:W
2205:P
2199:,
2194:U
2188:P
2182:(
2179:q
2159:W
2139:U
2117:W
2111:P
2086:U
2080:P
2047:1
2044:=
2037:2
2033:n
2027:|
2021:j
2017:V
2012:|
2007:|
2001:i
1997:V
1992:|
1983:k
1978:1
1975:=
1972:j
1962:k
1957:1
1954:=
1951:i
1938:2
1934:)
1928:j
1924:V
1920:,
1915:i
1911:V
1907:(
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1897:2
1893:n
1887:|
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1872:|
1867:|
1861:i
1857:V
1852:|
1843:k
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1835:=
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1814:=
1811:i
1803:=
1800:)
1795:P
1790:(
1787:q
1758:2
1754:)
1748:j
1744:V
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1727:(
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1615:j
1611:V
1607:,
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1538:P
1533:(
1530:q
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1497:,
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1450:V
1436:}
1431:k
1427:V
1423:,
1417:,
1412:1
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1401:=
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1371:)
1366:j
1362:W
1358:,
1353:i
1349:U
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1342:q
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1332:1
1329:=
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1316:k
1311:1
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1289:W
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1244:W
1230:}
1225:l
1221:W
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1206:1
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1198:{
1195:=
1190:W
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1157:}
1152:k
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1133:1
1129:U
1125:{
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1052:n
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1015:)
1012:W
1009:,
1006:U
1003:(
1000:q
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979:(
961:n
958:=
954:|
950:V
946:|
934:V
928:W
924:U
900:2
896:L
844:m
839:ε
834:M
824:m
814:M
804:k
798:G
793:ε
788:M
784:k
780:m
774:k
768:M
762:G
756:M
750:m
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707:|
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678:|
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547:}
542:k
538:V
534:,
528:,
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515:{
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485:k
465:V
448:ε
444:ε
436:ε
432:ε
414:.
404:|
400:)
397:B
394:,
391:A
388:(
385:d
379:)
376:Y
373:,
370:X
367:(
364:d
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346:|
344:Y
340:B
338:|
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332:X
328:A
326:|
321:Y
317:B
311:X
307:A
300:ε
294:Y
288:X
272:ε
258:.
255:Y
249:X
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242:Y
238:X
236:(
234:E
212:|
208:Y
204:|
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195:X
191:|
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150:Y
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144:X
141:(
138:d
125:)
123:Y
119:X
117:(
108:V
102:Y
98:X
85:V
75:G
70:ε
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