587:(partition hypergraphs into pseudorandom blocks) and a counting lemma (estimate the number of hypergraphs in an appropriate pseudorandom block). The key difficulty in the proof is to define the correct notion of hypergraph regularity. There were multiple attempts to define "partition" and "pseudorandom (regular) blocks" in a hypergraph, but none of them are able to give a strong counting lemma. The first correct definition of Szemerédi's regularity lemma for general hypergraphs is given by Rödl et al.
1507:
3925:
This method usually does not give a good quantitative bound, since the hidden constants in hypergraph removal lemma involves the inverse
Ackermann function. For a better quantitive bound, Leng, Sah, and Sawhney proved that
1309:
4016:
1301:
1044:
1140:
904:
2951:
3102:
3311:
2649:
3519:
136:
3188:
2716:
1736:
1202:
3390:
2540:
2331:
2214:
2485:
2097:
4392:
3875:
83:
2874:
385:
4147:
3231:
3775:
2052:
arithmetic progression, then use graph removal lemma to show that this graph cannot have too many hyperedges, which in turn shows that the original subset cannot be too big.
1935:
295:
2983:
2572:
2010:
4256:
4173:
2276:
1665:
3010:
690:
473:
4288:
3561:
2159:
29:
5124:
Bergelson, Vitaly; Leibman, Alexander; Ziegler, Tamar (February 2011). "The shifted primes and the multidimensional Szemerédi and polynomial Van der
Waerden theorems".
3920:
1897:
1776:
1550:
945:
569:
4089:
1809:
1641:
1603:
3718:
618:
4043:
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806:
774:
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1841:
4308:
4193:
4118:
4063:
3798:
3681:
3661:
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3621:
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3430:
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3142:
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3030:
2828:
2808:
2756:
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2505:
2426:
2406:
2354:
2296:
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2234:
2179:
2117:
2050:
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1955:
1570:
742:
658:
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533:
513:
493:
428:
405:
355:
335:
315:
256:
236:
216:
196:
176:
156:
24:
contains few copies of a given sub-hypergraph, then all of the copies can be eliminated by removing a small number of hyperedges. It is a generalization of the
816:
For example, we demonstrate an informal 3-hypergraph version of Szemerédi's regularity lemma, first given by Frankl and Rödl. Consider a partition of edges
4100:
The hypergraph removal lemma is used to prove the multidimensional Szemerédi theorem by J. Solymosi. The statement is that any for any finite subset
1502:{\displaystyle \left|d\left(G_{i}^{(2)},G_{j}^{(2)},G_{k}^{(2)}\right)-d\left(A_{i}^{(2)},A_{j}^{(2)},A_{k}^{(2)}\right)\right|\leq \varepsilon ,}
4397:
It is also used to prove the polynomial Szemerédi theorem, the finite field Szemerédi theorem and the finite abelian group Szemerédi theorem.
1851:
After proving the hypergraph regularity lemma, we can prove a hypergraph counting lemma. The rest of proof proceeds similarly to that of
1647:
We then subsequently define a regular partition as a partition in which the triples of parts that are not regular constitute at most an
4417:
3929:
1207:
950:
1049:
819:
2879:
3035:
3239:
2577:
3435:
88:
5175:
1844:
591:
584:
3147:
2654:
1673:
1145:
4821:
3336:
2510:
2301:
2184:
4570:
2431:
2058:
4317:
3803:
1738:
via a partition of the vertex set. As a result, we have the total data of hypergraph regularity as follows:
808:-hyperedges to regulate them, etc. In the end, we will reach a complex structure of regulating hyperedges.
50:
2833:
2032:. The high level idea of the proof is that, we construct a hypergraph from a subset without any length
594:, the partitions are performed on vertices (1-hyperedge) to regulate edges (2-hyperedge). However, for
360:
4123:
3193:
4893:
Czygrinow, Andrzej; Rödl, Vojtech (January 2000). "An
Algorithmic Regularity Lemma for Hypergraphs".
3723:
1902:
261:
4412:
1958:
36:
2956:
2545:
1964:
4511:
Gowers, William (2007-11-01). "Hypergraph regularity and the multidimensional Szemerédi theorem".
4231:
4152:
2259:
1650:
2988:
663:
433:
4261:
3524:
2122:
5068:
Leng, James; Sah, Ashwin; Sawhney, Mehtaab (2024). "Improved Bounds for Szemerédi's
Theorem".
3880:
1866:
1745:
1514:
909:
538:
5223:
4858:
Frieze, Alan; Kannan, Ravi (1999-02-01). "Quick
Approximation to Matrices and Applications".
4068:
1781:
1608:
1575:
3686:
597:
5218:
4620:
4530:
4457:
4021:
2761:
2359:
779:
747:
695:
4198:
1817:
8:
4407:
1852:
580:
25:
4998:
Nagle, Brendan; Rödl, Vojtěch (2003-07-17). "Regularity properties for triple systems".
4928:
Chung, Fan R.K. (2007-07-05). "Regularity lemmas for hypergraphs and quasi-randomness".
4624:
4534:
4461:
32:. It was first proved by Nagle, Rödl, Schacht and Skokan and, independently, by Gowers.
5133:
5069:
4797:
4520:
4488:
4445:
4293:
4178:
4103:
4048:
3783:
3666:
3646:
3626:
3606:
3586:
3566:
3415:
3395:
3316:
3127:
3107:
3015:
2813:
2793:
2741:
2721:
2490:
2411:
2391:
2339:
2281:
2239:
2219:
2164:
2102:
2035:
2015:
1940:
1555:
727:
643:
623:
518:
498:
478:
413:
390:
340:
320:
300:
241:
221:
201:
181:
161:
141:
4743:
4710:
4651:
4608:
5194:
5151:
5106:
5050:
5015:
4980:
4945:
4910:
4875:
4840:
4789:
4748:
4730:
4691:
4656:
4638:
4589:
4584:
4565:
4546:
4493:
4475:
5184:
5143:
5098:
5042:
5007:
4972:
4937:
4902:
4867:
4830:
4779:
4738:
4722:
4683:
4646:
4628:
4579:
4538:
4483:
4465:
4311:
4542:
1843:
such that the graphs in (1) are extremely pseudorandom (in a fashion resembling
5147:
5033:
Frankl, Peter; Rödl, Vojtěch (2002-02-07). "Extremal problems on set systems".
5102:
4963:
Frankl, P.; Rödl, V. (December 1992). "The
Uniformity Lemma for hypergraphs".
4906:
5212:
5198:
5155:
5110:
5054:
5019:
4984:
4949:
4914:
4879:
4844:
4793:
4734:
4695:
4642:
4593:
4550:
4479:
4633:
4470:
4941:
4835:
4816:
4752:
4726:
4687:
4660:
4497:
4444:
Rodl, V.; Nagle, B.; Skokan, J.; Schacht, M.; Kohayakawa, Y. (2005-05-26).
4871:
692:, and fail to find a counting lemma. The correct version has to partition
5189:
5170:
5089:
SOLYMOSI, J. (March 2004). "A Note on a
Question of Erdős and Graham".
4976:
4801:
4446:"From The Cover: The hypergraph regularity method and its applications"
28:. The special case in which the graph is a tetrahedron is known as the
21:
5046:
5011:
4815:
Kohayakawa, Yoshiharu; Rödl, Vojtěch; Skokan, Jozef (February 2002).
4784:
4767:
640:-hyperedges using only 1-hyperedge, we will lose information of all
5074:
5138:
4525:
35:
The hypergraph removal lemma can be used to prove results such as
4011:{\displaystyle |A|\leq {\frac {N}{\exp(-(\log \log N)^{c_{k}})}}}
1296:{\displaystyle \left(A_{i}^{(2)},A_{j}^{(2)},A_{k}^{(2)}\right),}
1039:{\displaystyle \left(G_{i}^{(2)},G_{j}^{(2)},G_{k}^{(2)}\right).}
4674:
Chung, Fan R. K. (1990). "Quasi-random classes of hypergraphs".
1135:{\displaystyle \left(G_{i}^{(2)},G_{j}^{(2)},G_{k}^{(2)}\right)}
899:{\displaystyle E(K_{n})=G_{1}^{(2)}\cup \dots \cup G_{l}^{(2)}}
4817:"Hypergraphs, Quasi-randomness, and Conditions for Regularity"
574:
3521:. Therefore, by the hypergraph removal lemma, we can remove
2946:{\displaystyle \alpha _{i}=\sum _{j\neq i}(j-i)v_{j}\in A}
3097:{\displaystyle \alpha _{i+1}-\alpha _{i}=-\sum _{j}v_{j}}
5123:
3306:{\displaystyle (v_{j}\in V_{j})_{j\in \setminus \{i\}}}
2644:{\displaystyle (v_{j}\in V_{j})_{j\in \setminus \{i\}}}
811:
776:-hyperedges, we can go a level deeper and partition on
579:
The high level idea of the proof is similar to that of
4443:
3514:{\displaystyle {\frac {1}{k}}e(G)=O(N^{k-1})=o(N^{k})}
1142:
is "pseudorandom" in the sense that for all subgraphs
131:{\displaystyle \delta =\delta (\varepsilon ,r,m)>0}
5168:
4320:
4296:
4264:
4234:
4201:
4181:
4155:
4126:
4106:
4071:
4051:
4024:
3932:
3883:
3806:
3786:
3726:
3689:
3669:
3649:
3629:
3609:
3589:
3569:
3527:
3438:
3418:
3398:
3339:
3319:
3242:
3196:
3150:
3130:
3110:
3038:
3018:
2991:
2959:
2882:
2836:
2816:
2796:
2764:
2744:
2724:
2657:
2580:
2548:
2513:
2493:
2434:
2414:
2394:
2362:
2342:
2304:
2284:
2262:
2242:
2222:
2187:
2167:
2125:
2105:
2061:
2038:
2018:
1967:
1943:
1905:
1869:
1820:
1784:
1748:
1676:
1653:
1611:
1578:
1558:
1517:
1312:
1210:
1148:
1052:
953:
912:
822:
782:
750:
730:
698:
666:
646:
626:
600:
541:
521:
501:
481:
436:
416:
393:
363:
343:
323:
303:
264:
244:
224:
204:
184:
164:
144:
91:
53:
4814:
1670:In addition to this, we need to further regularize
1667:fraction of all triples of parts in the partition.
4386:
4302:
4282:
4250:
4220:
4187:
4167:
4141:
4112:
4083:
4057:
4037:
4010:
3914:
3869:
3792:
3769:
3712:
3675:
3655:
3635:
3615:
3595:
3575:
3555:
3513:
3424:
3404:
3384:
3325:
3305:
3225:
3182:
3136:
3116:
3096:
3024:
3004:
2977:
2945:
2868:
2822:
2802:
2782:
2750:
2730:
2710:
2643:
2566:
2534:
2499:
2479:
2420:
2400:
2380:
2348:
2325:
2290:
2270:
2248:
2228:
2208:
2173:
2153:
2111:
2091:
2044:
2024:
2004:
1949:
1929:
1891:
1835:
1803:
1770:
1730:
1659:
1635:
1597:
1564:
1544:
1501:
1295:
1196:
1134:
1038:
939:
898:
800:
768:
736:
716:
684:
652:
632:
612:
563:
527:
507:
487:
467:
422:
399:
379:
349:
329:
309:
289:
250:
230:
210:
190:
170:
150:
130:
77:
5169:Furstenberg, H.; Katznelson, Y. (December 1991).
3144:arithmetic progression, it must be the case that
410:An equivalent formulation is that, for any graph
317:, then it is possible to eliminate all copies of
47:The hypergraph removal lemma states that for any
5210:
4563:
5171:"A density version of the Hales-Jewett theorem"
4766:Chung, F. R. K.; Graham, R. L. (January 1991).
4613:Proceedings of the National Academy of Sciences
4450:Proceedings of the National Academy of Sciences
3183:{\displaystyle \alpha _{1}=\cdots =\alpha _{k}}
2711:{\displaystyle \sum _{j\neq i}(j-i)v_{j}\in A.}
1858:
5067:
4564:Haviland, Julie; Thomason, Andrew (May 1989).
3032:arithmetic progression with common difference
1731:{\displaystyle G_{1}^{(2)},\dots ,G_{l}^{(2)}}
1197:{\displaystyle A_{i}^{(2)}\subset G_{i}^{(2)}}
4892:
4607:Chung, F. R. K.; Graham, R. L. (1989-11-01).
2099:be a subset that does not contain any length
39:and the multi-dimensional Szemerédi theorem.
4857:
4772:Journal of the American Mathematical Society
4381:
4327:
4290:, that is, a dilated and translated copy of
3298:
3292:
2636:
2630:
2086:
2068:
1924:
1906:
4765:
4708:
4606:
3385:{\displaystyle v_{i}=-\sum _{j\neq i}v_{j}}
2161:be a large enough integer. We can think of
5032:
4962:
4418:Problems involving arithmetic progressions
2535:{\displaystyle \mathbb {Z} /M\mathbb {Z} }
2326:{\displaystyle \mathbb {Z} /M\mathbb {Z} }
2209:{\displaystyle \mathbb {Z} /M\mathbb {Z} }
575:Proof idea of the hypergraph removal lemma
5188:
5137:
5073:
4997:
4834:
4783:
4742:
4650:
4632:
4583:
4524:
4487:
4469:
4129:
2528:
2515:
2480:{\displaystyle V_{1},V_{2},\ldots ,V_{k}}
2319:
2306:
2264:
2202:
2189:
744:-hyperedges. To gain more control of the
5091:Combinatorics, Probability and Computing
5088:
2092:{\displaystyle A\subset \{1,\ldots ,N\}}
4709:Chung, F. R. K.; Graham, R. L. (1990).
4387:{\displaystyle S=\{(0,0),(0,1),(1,0)\}}
947:there are a lot of triangles on top of
5211:
4510:
3870:{\displaystyle kM^{k-2}|A|=o(N^{k-1})}
4927:
4673:
1899:be the size of the largest subset of
1204:with not too few triangles on top of
78:{\displaystyle \varepsilon ,r,m>0}
4439:
4437:
4435:
4433:
2574:, we add a hyperedge among vertices
812:Proof idea for 3-uniform hypergraphs
2869:{\displaystyle v_{1},\ldots ,v_{k}}
583:. We prove a hypergraph version of
198:vertices the following is true: if
13:
5035:Random Structures & Algorithms
5000:Random Structures & Algorithms
4930:Random Structures & Algorithms
3333:that this edge lies in by finding
14:
5235:
4430:
3563:edges to eliminate all copies of
3289:
2627:
724:-hyperedges in order to regulate
495:, we can eliminate all copies of
380:{\displaystyle \varepsilon n^{r}}
258:-uniform hypergraph with at most
4715:Random Structures and Algorithms
4676:Random Structures and Algorithms
4142:{\displaystyle \mathbb {Z} ^{r}}
3226:{\displaystyle \sum _{j}v_{j}=0}
660:-hyperedges in the middle where
5162:
5117:
5082:
5061:
5026:
4991:
4956:
4921:
4886:
4822:Journal of Combinatorial Theory
4094:
3770:{\displaystyle e(G)=o(N^{k-1})}
3313:, we can find a unique copy of
2810:contains an isomorphic copy of
2507:element vertex sets indexed by
1937:that does not contain a length
1930:{\displaystyle \{1,\ldots ,N\}}
290:{\displaystyle \delta n^{v(H)}}
5176:Journal d'Analyse Mathématique
4851:
4808:
4759:
4702:
4667:
4600:
4557:
4504:
4378:
4366:
4360:
4348:
4342:
4330:
4258:contains a subset of the form
4209:
4202:
4002:
3986:
3967:
3961:
3942:
3934:
3909:
3903:
3893:
3885:
3864:
3845:
3835:
3827:
3764:
3745:
3736:
3730:
3699:
3693:
3550:
3531:
3508:
3495:
3486:
3467:
3458:
3452:
3286:
3280:
3270:
3243:
2924:
2912:
2777:
2765:
2686:
2674:
2624:
2618:
2608:
2581:
2375:
2363:
2278:, it also doesn't have length
1999:
1993:
1984:
1978:
1886:
1880:
1830:
1824:
1796:
1790:
1765:
1752:
1723:
1717:
1693:
1687:
1630:
1612:
1605:among all triangles on top of
1590:
1584:
1539:
1521:
1475:
1469:
1451:
1445:
1427:
1421:
1390:
1384:
1366:
1360:
1342:
1336:
1280:
1274:
1256:
1250:
1232:
1226:
1189:
1183:
1165:
1159:
1122:
1116:
1098:
1092:
1074:
1068:
1023:
1017:
999:
993:
975:
969:
931:
913:
891:
885:
861:
855:
839:
826:
795:
783:
763:
751:
711:
699:
558:
545:
462:
457:
451:
440:
282:
276:
119:
101:
1:
4423:
3683:, we need to remove at least
3643:, to eliminate all copies of
2978:{\displaystyle 1\leq i\leq j}
2567:{\displaystyle 1\leq i\leq k}
2005:{\displaystyle r_{k}(N)=o(N)}
4585:10.1016/0012-365x(89)90093-9
4251:{\displaystyle \delta n^{r}}
4195:large enough, any subset of
4168:{\displaystyle \delta >0}
3780:The number of hyperedges in
2271:{\displaystyle \mathbb {Z} }
2119:arithmetic progression. Let
1859:Proof of Szemerédi's theorem
1845:Szemerédi's regularity lemma
1660:{\displaystyle \varepsilon }
592:Szemerédi's regularity lemma
585:Szemerédi's regularity lemma
42:
7:
5126:Comptes Rendus Mathématique
4566:"Pseudo-random hypergraphs"
4543:10.4007/annals.2007.166.897
4401:
4065:. It is the best bound for
3603:. Since every hyperedge of
3005:{\displaystyle \alpha _{i}}
1811:sits pseudorandomly on top;
906:such that for most triples
685:{\displaystyle 1<j<k}
468:{\displaystyle o(n^{v(H)})}
10:
5240:
5148:10.1016/j.crma.2010.11.028
4768:"Quasi-Random Set Systems"
4711:"Quasi-random hypergraphs"
4609:"Quasi-random hypergraphs"
4283:{\displaystyle a\cdot S+d}
3556:{\displaystyle o(N^{k-1})}
3392:. The number of copies of
2298:arithmetic progression in
2256:arithmetic progression in
2154:{\displaystyle M=k^{2}N+1}
1552:denotes the proportion of
5103:10.1017/s0963548303005959
4907:10.1137/s0097539799351729
4895:SIAM Journal on Computing
3236:Thus, for each hyperedge
30:tetrahedron removal lemma
4965:Graphs and Combinatorics
3915:{\displaystyle |A|=o(N)}
2790:-uniform hypergraph. If
1957:arithmetic progression.
1892:{\displaystyle r_{k}(N)}
1771:{\displaystyle E(K_{n})}
1545:{\displaystyle d(X,Y,Z)}
940:{\displaystyle (i,j,k),}
620:, if we simply regulate
564:{\displaystyle o(n^{r})}
297:subgraphs isomorphic to
18:hypergraph removal lemma
4634:10.1073/pnas.86.21.8175
4471:10.1073/pnas.0502771102
4314:is a special case when
4084:{\displaystyle k\geq 5}
3877:, which concludes that
3623:is in a unique copy of
1804:{\displaystyle G^{(3)}}
1636:{\displaystyle (X,Y,Z)}
1598:{\displaystyle G^{(3)}}
4942:10.1002/rsa.3240020208
4836:10.1006/jcta.2001.3217
4727:10.1002/rsa.3240010108
4688:10.1002/rsa.3240010401
4388:
4304:
4284:
4252:
4222:
4189:
4169:
4143:
4114:
4085:
4059:
4039:
4012:
3916:
3871:
3794:
3771:
3714:
3713:{\displaystyle e(G)/k}
3677:
3657:
3637:
3617:
3597:
3577:
3557:
3515:
3426:
3406:
3386:
3327:
3307:
3227:
3184:
3138:
3118:
3098:
3026:
3006:
2979:
2947:
2870:
2824:
2804:
2784:
2752:
2732:
2712:
2645:
2568:
2536:
2501:
2481:
2422:
2402:
2382:
2350:
2327:
2292:
2272:
2250:
2230:
2210:
2175:
2155:
2113:
2093:
2046:
2026:
2006:
1951:
1931:
1893:
1837:
1805:
1778:into graphs such that
1772:
1732:
1661:
1637:
1599:
1572:-uniform hyperedge in
1566:
1546:
1503:
1297:
1198:
1136:
1040:
941:
900:
802:
770:
738:
718:
686:
654:
634:
614:
613:{\displaystyle k>2}
565:
529:
509:
489:
469:
424:
401:
381:
351:
331:
311:
291:
252:
232:
212:
192:
172:
152:
132:
79:
4872:10.1007/s004930050052
4513:Annals of Mathematics
4389:
4305:
4285:
4253:
4223:
4190:
4170:
4144:
4115:
4086:
4060:
4040:
4038:{\displaystyle c_{k}}
4013:
3917:
3872:
3795:
3772:
3715:
3678:
3658:
3638:
3618:
3598:
3578:
3558:
3516:
3427:
3407:
3387:
3328:
3308:
3228:
3185:
3139:
3119:
3099:
3027:
3007:
2985:. However, note that
2980:
2948:
2871:
2825:
2805:
2785:
2783:{\displaystyle (k-1)}
2753:
2733:
2713:
2646:
2569:
2537:
2502:
2482:
2423:
2403:
2383:
2381:{\displaystyle (k-1)}
2351:
2328:
2293:
2273:
2251:
2231:
2211:
2176:
2156:
2114:
2094:
2047:
2027:
2007:
1952:
1932:
1894:
1838:
1806:
1773:
1733:
1662:
1638:
1600:
1567:
1547:
1504:
1298:
1199:
1137:
1041:
942:
901:
803:
801:{\displaystyle (k-2)}
771:
769:{\displaystyle (k-1)}
739:
719:
717:{\displaystyle (k-1)}
687:
655:
635:
615:
566:
530:
510:
490:
470:
425:
402:
382:
352:
332:
312:
292:
253:
233:
213:
193:
173:
153:
133:
80:
16:In graph theory, the
4571:Discrete Mathematics
4318:
4294:
4262:
4232:
4221:{\displaystyle ^{r}}
4199:
4179:
4153:
4124:
4104:
4069:
4049:
4022:
3930:
3881:
3804:
3784:
3724:
3687:
3667:
3647:
3627:
3607:
3587:
3567:
3525:
3436:
3416:
3396:
3337:
3317:
3240:
3194:
3148:
3128:
3108:
3036:
3016:
2989:
2957:
2880:
2834:
2814:
2794:
2762:
2742:
2722:
2655:
2578:
2546:
2511:
2491:
2432:
2412:
2392:
2388:-uniform hypergraph
2360:
2340:
2336:We will construct a
2302:
2282:
2260:
2240:
2236:doesn't have length
2220:
2185:
2165:
2123:
2103:
2059:
2036:
2016:
1965:
1941:
1903:
1867:
1836:{\displaystyle V(G)}
1818:
1782:
1746:
1674:
1651:
1609:
1576:
1556:
1515:
1310:
1208:
1146:
1050:
951:
910:
820:
780:
748:
728:
696:
664:
644:
624:
598:
539:
519:
499:
479:
434:
414:
391:
361:
357:by removing at most
341:
321:
301:
262:
242:
222:
202:
182:
162:
158:-uniform hypergraph
142:
89:
51:
4625:1989PNAS...86.8175C
4535:2007arXiv0710.3032G
4462:2005PNAS..102.8109R
4413:Szemerédi's theorem
4408:Graph removal lemma
2487:, all of which are
1959:Szemerédi's theorem
1853:Graph removal lemma
1727:
1697:
1479:
1455:
1431:
1394:
1370:
1346:
1284:
1260:
1236:
1193:
1169:
1126:
1102:
1078:
1027:
1003:
979:
895:
865:
581:graph removal lemma
37:Szemerédi's theorem
26:graph removal lemma
20:states that when a
5190:10.1007/bf03041066
4977:10.1007/bf02351586
4384:
4300:
4280:
4248:
4218:
4185:
4165:
4139:
4110:
4081:
4055:
4035:
4018:for some constant
4008:
3912:
3867:
3790:
3767:
3710:
3673:
3653:
3633:
3613:
3593:
3573:
3553:
3511:
3422:
3402:
3382:
3371:
3323:
3303:
3223:
3206:
3180:
3134:
3114:
3094:
3083:
3022:
3002:
2975:
2943:
2911:
2866:
2820:
2800:
2780:
2748:
2728:
2708:
2673:
2641:
2564:
2532:
2497:
2477:
2418:
2398:
2378:
2346:
2323:
2288:
2268:
2246:
2226:
2206:
2171:
2151:
2109:
2089:
2042:
2022:
2002:
1947:
1927:
1889:
1833:
1801:
1768:
1728:
1707:
1677:
1657:
1633:
1595:
1562:
1542:
1499:
1459:
1435:
1411:
1374:
1350:
1326:
1293:
1264:
1240:
1216:
1194:
1173:
1149:
1132:
1106:
1082:
1058:
1036:
1007:
983:
959:
937:
896:
875:
845:
798:
766:
734:
714:
682:
650:
630:
610:
561:
525:
505:
485:
465:
420:
397:
377:
347:
327:
307:
287:
248:
228:
208:
188:
168:
148:
138:such that for any
128:
75:
5047:10.1002/rsa.10017
5012:10.1002/rsa.10094
4619:(21): 8175–8177.
4456:(23): 8109–8113.
4303:{\displaystyle S}
4228:of size at least
4188:{\displaystyle n}
4113:{\displaystyle S}
4058:{\displaystyle k}
4006:
3793:{\displaystyle G}
3676:{\displaystyle G}
3656:{\displaystyle H}
3636:{\displaystyle H}
3616:{\displaystyle G}
3596:{\displaystyle G}
3576:{\displaystyle H}
3447:
3425:{\displaystyle G}
3405:{\displaystyle H}
3356:
3326:{\displaystyle H}
3197:
3137:{\displaystyle k}
3117:{\displaystyle A}
3074:
3025:{\displaystyle k}
2896:
2823:{\displaystyle H}
2803:{\displaystyle G}
2751:{\displaystyle k}
2731:{\displaystyle H}
2658:
2500:{\displaystyle M}
2421:{\displaystyle A}
2401:{\displaystyle G}
2349:{\displaystyle k}
2291:{\displaystyle k}
2249:{\displaystyle k}
2229:{\displaystyle A}
2174:{\displaystyle A}
2112:{\displaystyle k}
2045:{\displaystyle k}
2025:{\displaystyle k}
2012:for any constant
1950:{\displaystyle k}
1565:{\displaystyle 3}
737:{\displaystyle k}
653:{\displaystyle j}
633:{\displaystyle k}
528:{\displaystyle G}
508:{\displaystyle H}
488:{\displaystyle H}
423:{\displaystyle G}
400:{\displaystyle G}
350:{\displaystyle G}
330:{\displaystyle H}
310:{\displaystyle H}
251:{\displaystyle r}
231:{\displaystyle n}
211:{\displaystyle G}
191:{\displaystyle m}
171:{\displaystyle H}
151:{\displaystyle r}
5231:
5203:
5202:
5192:
5166:
5160:
5159:
5141:
5132:(3–4): 123–125.
5121:
5115:
5114:
5086:
5080:
5079:
5077:
5065:
5059:
5058:
5030:
5024:
5023:
4995:
4989:
4988:
4960:
4954:
4953:
4925:
4919:
4918:
4901:(4): 1041–1066.
4890:
4884:
4883:
4855:
4849:
4848:
4838:
4812:
4806:
4805:
4787:
4763:
4757:
4756:
4746:
4706:
4700:
4699:
4671:
4665:
4664:
4654:
4636:
4604:
4598:
4597:
4587:
4578:(1–3): 255–278.
4561:
4555:
4554:
4528:
4508:
4502:
4501:
4491:
4473:
4441:
4393:
4391:
4390:
4385:
4309:
4307:
4306:
4301:
4289:
4287:
4286:
4281:
4257:
4255:
4254:
4249:
4247:
4246:
4227:
4225:
4224:
4219:
4217:
4216:
4194:
4192:
4191:
4186:
4174:
4172:
4171:
4166:
4148:
4146:
4145:
4140:
4138:
4137:
4132:
4119:
4117:
4116:
4111:
4090:
4088:
4087:
4082:
4064:
4062:
4061:
4056:
4044:
4042:
4041:
4036:
4034:
4033:
4017:
4015:
4014:
4009:
4007:
4005:
4001:
4000:
3999:
3998:
3950:
3945:
3937:
3921:
3919:
3918:
3913:
3896:
3888:
3876:
3874:
3873:
3868:
3863:
3862:
3838:
3830:
3825:
3824:
3799:
3797:
3796:
3791:
3776:
3774:
3773:
3768:
3763:
3762:
3719:
3717:
3716:
3711:
3706:
3682:
3680:
3679:
3674:
3662:
3660:
3659:
3654:
3642:
3640:
3639:
3634:
3622:
3620:
3619:
3614:
3602:
3600:
3599:
3594:
3582:
3580:
3579:
3574:
3562:
3560:
3559:
3554:
3549:
3548:
3520:
3518:
3517:
3512:
3507:
3506:
3485:
3484:
3448:
3440:
3431:
3429:
3428:
3423:
3411:
3409:
3408:
3403:
3391:
3389:
3388:
3383:
3381:
3380:
3370:
3349:
3348:
3332:
3330:
3329:
3324:
3312:
3310:
3309:
3304:
3302:
3301:
3268:
3267:
3255:
3254:
3232:
3230:
3229:
3224:
3216:
3215:
3205:
3189:
3187:
3186:
3181:
3179:
3178:
3160:
3159:
3143:
3141:
3140:
3135:
3123:
3121:
3120:
3115:
3103:
3101:
3100:
3095:
3093:
3092:
3082:
3067:
3066:
3054:
3053:
3031:
3029:
3028:
3023:
3011:
3009:
3008:
3003:
3001:
3000:
2984:
2982:
2981:
2976:
2952:
2950:
2949:
2944:
2936:
2935:
2910:
2892:
2891:
2875:
2873:
2872:
2867:
2865:
2864:
2846:
2845:
2829:
2827:
2826:
2821:
2809:
2807:
2806:
2801:
2789:
2787:
2786:
2781:
2757:
2755:
2754:
2749:
2738:be the complete
2737:
2735:
2734:
2729:
2717:
2715:
2714:
2709:
2698:
2697:
2672:
2650:
2648:
2647:
2642:
2640:
2639:
2606:
2605:
2593:
2592:
2573:
2571:
2570:
2565:
2541:
2539:
2538:
2533:
2531:
2523:
2518:
2506:
2504:
2503:
2498:
2486:
2484:
2483:
2478:
2476:
2475:
2457:
2456:
2444:
2443:
2427:
2425:
2424:
2419:
2407:
2405:
2404:
2399:
2387:
2385:
2384:
2379:
2355:
2353:
2352:
2347:
2332:
2330:
2329:
2324:
2322:
2314:
2309:
2297:
2295:
2294:
2289:
2277:
2275:
2274:
2269:
2267:
2255:
2253:
2252:
2247:
2235:
2233:
2232:
2227:
2215:
2213:
2212:
2207:
2205:
2197:
2192:
2180:
2178:
2177:
2172:
2160:
2158:
2157:
2152:
2141:
2140:
2118:
2116:
2115:
2110:
2098:
2096:
2095:
2090:
2051:
2049:
2048:
2043:
2031:
2029:
2028:
2023:
2011:
2009:
2008:
2003:
1977:
1976:
1956:
1954:
1953:
1948:
1936:
1934:
1933:
1928:
1898:
1896:
1895:
1890:
1879:
1878:
1842:
1840:
1839:
1834:
1810:
1808:
1807:
1802:
1800:
1799:
1777:
1775:
1774:
1769:
1764:
1763:
1737:
1735:
1734:
1729:
1726:
1715:
1696:
1685:
1666:
1664:
1663:
1658:
1642:
1640:
1639:
1634:
1604:
1602:
1601:
1596:
1594:
1593:
1571:
1569:
1568:
1563:
1551:
1549:
1548:
1543:
1508:
1506:
1505:
1500:
1489:
1485:
1484:
1480:
1478:
1467:
1454:
1443:
1430:
1419:
1399:
1395:
1393:
1382:
1369:
1358:
1345:
1334:
1302:
1300:
1299:
1294:
1289:
1285:
1283:
1272:
1259:
1248:
1235:
1224:
1203:
1201:
1200:
1195:
1192:
1181:
1168:
1157:
1141:
1139:
1138:
1133:
1131:
1127:
1125:
1114:
1101:
1090:
1077:
1066:
1045:
1043:
1042:
1037:
1032:
1028:
1026:
1015:
1002:
991:
978:
967:
946:
944:
943:
938:
905:
903:
902:
897:
894:
883:
864:
853:
838:
837:
807:
805:
804:
799:
775:
773:
772:
767:
743:
741:
740:
735:
723:
721:
720:
715:
691:
689:
688:
683:
659:
657:
656:
651:
639:
637:
636:
631:
619:
617:
616:
611:
570:
568:
567:
562:
557:
556:
534:
532:
531:
526:
514:
512:
511:
506:
494:
492:
491:
486:
474:
472:
471:
466:
461:
460:
429:
427:
426:
421:
406:
404:
403:
398:
387:hyperedges from
386:
384:
383:
378:
376:
375:
356:
354:
353:
348:
336:
334:
333:
328:
316:
314:
313:
308:
296:
294:
293:
288:
286:
285:
257:
255:
254:
249:
237:
235:
234:
229:
217:
215:
214:
209:
197:
195:
194:
189:
177:
175:
174:
169:
157:
155:
154:
149:
137:
135:
134:
129:
84:
82:
81:
76:
5239:
5238:
5234:
5233:
5232:
5230:
5229:
5228:
5209:
5208:
5207:
5206:
5167:
5163:
5122:
5118:
5087:
5083:
5066:
5062:
5031:
5027:
4996:
4992:
4961:
4957:
4926:
4922:
4891:
4887:
4856:
4852:
4813:
4809:
4785:10.2307/2939258
4764:
4760:
4707:
4703:
4672:
4668:
4605:
4601:
4562:
4558:
4509:
4505:
4442:
4431:
4426:
4404:
4319:
4316:
4315:
4312:Corners theorem
4295:
4292:
4291:
4263:
4260:
4259:
4242:
4238:
4233:
4230:
4229:
4212:
4208:
4200:
4197:
4196:
4180:
4177:
4176:
4154:
4151:
4150:
4133:
4128:
4127:
4125:
4122:
4121:
4105:
4102:
4101:
4097:
4070:
4067:
4066:
4050:
4047:
4046:
4029:
4025:
4023:
4020:
4019:
3994:
3990:
3989:
3985:
3954:
3949:
3941:
3933:
3931:
3928:
3927:
3892:
3884:
3882:
3879:
3878:
3852:
3848:
3834:
3826:
3814:
3810:
3805:
3802:
3801:
3785:
3782:
3781:
3752:
3748:
3725:
3722:
3721:
3702:
3688:
3685:
3684:
3668:
3665:
3664:
3648:
3645:
3644:
3628:
3625:
3624:
3608:
3605:
3604:
3588:
3585:
3584:
3568:
3565:
3564:
3538:
3534:
3526:
3523:
3522:
3502:
3498:
3474:
3470:
3439:
3437:
3434:
3433:
3417:
3414:
3413:
3397:
3394:
3393:
3376:
3372:
3360:
3344:
3340:
3338:
3335:
3334:
3318:
3315:
3314:
3273:
3269:
3263:
3259:
3250:
3246:
3241:
3238:
3237:
3211:
3207:
3201:
3195:
3192:
3191:
3174:
3170:
3155:
3151:
3149:
3146:
3145:
3129:
3126:
3125:
3109:
3106:
3105:
3088:
3084:
3078:
3062:
3058:
3043:
3039:
3037:
3034:
3033:
3017:
3014:
3013:
2996:
2992:
2990:
2987:
2986:
2958:
2955:
2954:
2931:
2927:
2900:
2887:
2883:
2881:
2878:
2877:
2860:
2856:
2841:
2837:
2835:
2832:
2831:
2815:
2812:
2811:
2795:
2792:
2791:
2763:
2760:
2759:
2743:
2740:
2739:
2723:
2720:
2719:
2693:
2689:
2662:
2656:
2653:
2652:
2651:if and only if
2611:
2607:
2601:
2597:
2588:
2584:
2579:
2576:
2575:
2547:
2544:
2543:
2527:
2519:
2514:
2512:
2509:
2508:
2492:
2489:
2488:
2471:
2467:
2452:
2448:
2439:
2435:
2433:
2430:
2429:
2413:
2410:
2409:
2393:
2390:
2389:
2361:
2358:
2357:
2341:
2338:
2337:
2318:
2310:
2305:
2303:
2300:
2299:
2283:
2280:
2279:
2263:
2261:
2258:
2257:
2241:
2238:
2237:
2221:
2218:
2217:
2201:
2193:
2188:
2186:
2183:
2182:
2181:as a subset of
2166:
2163:
2162:
2136:
2132:
2124:
2121:
2120:
2104:
2101:
2100:
2060:
2057:
2056:
2037:
2034:
2033:
2017:
2014:
2013:
1972:
1968:
1966:
1963:
1962:
1942:
1939:
1938:
1904:
1901:
1900:
1874:
1870:
1868:
1865:
1864:
1861:
1819:
1816:
1815:
1814:a partition of
1789:
1785:
1783:
1780:
1779:
1759:
1755:
1747:
1744:
1743:
1742:a partition of
1716:
1711:
1686:
1681:
1675:
1672:
1671:
1652:
1649:
1648:
1610:
1607:
1606:
1583:
1579:
1577:
1574:
1573:
1557:
1554:
1553:
1516:
1513:
1512:
1468:
1463:
1444:
1439:
1420:
1415:
1410:
1406:
1383:
1378:
1359:
1354:
1335:
1330:
1325:
1321:
1317:
1313:
1311:
1308:
1307:
1273:
1268:
1249:
1244:
1225:
1220:
1215:
1211:
1209:
1206:
1205:
1182:
1177:
1158:
1153:
1147:
1144:
1143:
1115:
1110:
1091:
1086:
1067:
1062:
1057:
1053:
1051:
1048:
1047:
1016:
1011:
992:
987:
968:
963:
958:
954:
952:
949:
948:
911:
908:
907:
884:
879:
854:
849:
833:
829:
821:
818:
817:
814:
781:
778:
777:
749:
746:
745:
729:
726:
725:
697:
694:
693:
665:
662:
661:
645:
642:
641:
625:
622:
621:
599:
596:
595:
577:
552:
548:
540:
537:
536:
520:
517:
516:
500:
497:
496:
480:
477:
476:
447:
443:
435:
432:
431:
415:
412:
411:
392:
389:
388:
371:
367:
362:
359:
358:
342:
339:
338:
322:
319:
318:
302:
299:
298:
272:
268:
263:
260:
259:
243:
240:
239:
223:
220:
219:
203:
200:
199:
183:
180:
179:
163:
160:
159:
143:
140:
139:
90:
87:
86:
85:, there exists
52:
49:
48:
45:
12:
11:
5:
5237:
5227:
5226:
5221:
5205:
5204:
5161:
5116:
5097:(2): 263–267.
5081:
5060:
5041:(2): 131–164.
5025:
5006:(3): 264–332.
4990:
4971:(4): 309–312.
4955:
4936:(2): 241–252.
4920:
4885:
4866:(2): 175–220.
4850:
4829:(2): 307–352.
4807:
4758:
4721:(1): 105–124.
4701:
4682:(4): 363–382.
4666:
4599:
4556:
4519:(3): 897–946.
4503:
4428:
4427:
4425:
4422:
4421:
4420:
4415:
4410:
4403:
4400:
4399:
4398:
4395:
4383:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4359:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4335:
4332:
4329:
4326:
4323:
4299:
4279:
4276:
4273:
4270:
4267:
4245:
4241:
4237:
4215:
4211:
4207:
4204:
4184:
4164:
4161:
4158:
4136:
4131:
4109:
4096:
4093:
4080:
4077:
4074:
4054:
4032:
4028:
4004:
3997:
3993:
3988:
3984:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3960:
3957:
3953:
3948:
3944:
3940:
3936:
3911:
3908:
3905:
3902:
3899:
3895:
3891:
3887:
3866:
3861:
3858:
3855:
3851:
3847:
3844:
3841:
3837:
3833:
3829:
3823:
3820:
3817:
3813:
3809:
3789:
3766:
3761:
3758:
3755:
3751:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3709:
3705:
3701:
3698:
3695:
3692:
3672:
3652:
3632:
3612:
3592:
3572:
3552:
3547:
3544:
3541:
3537:
3533:
3530:
3510:
3505:
3501:
3497:
3494:
3491:
3488:
3483:
3480:
3477:
3473:
3469:
3466:
3463:
3460:
3457:
3454:
3451:
3446:
3443:
3421:
3401:
3379:
3375:
3369:
3366:
3363:
3359:
3355:
3352:
3347:
3343:
3322:
3300:
3297:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3272:
3266:
3262:
3258:
3253:
3249:
3245:
3222:
3219:
3214:
3210:
3204:
3200:
3177:
3173:
3169:
3166:
3163:
3158:
3154:
3133:
3124:has no length
3113:
3091:
3087:
3081:
3077:
3073:
3070:
3065:
3061:
3057:
3052:
3049:
3046:
3042:
3021:
2999:
2995:
2974:
2971:
2968:
2965:
2962:
2942:
2939:
2934:
2930:
2926:
2923:
2920:
2917:
2914:
2909:
2906:
2903:
2899:
2895:
2890:
2886:
2863:
2859:
2855:
2852:
2849:
2844:
2840:
2830:with vertices
2819:
2799:
2779:
2776:
2773:
2770:
2767:
2747:
2727:
2707:
2704:
2701:
2696:
2692:
2688:
2685:
2682:
2679:
2676:
2671:
2668:
2665:
2661:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2610:
2604:
2600:
2596:
2591:
2587:
2583:
2563:
2560:
2557:
2554:
2551:
2530:
2526:
2522:
2517:
2496:
2474:
2470:
2466:
2463:
2460:
2455:
2451:
2447:
2442:
2438:
2417:
2397:
2377:
2374:
2371:
2368:
2365:
2345:
2321:
2317:
2313:
2308:
2287:
2266:
2245:
2225:
2216:. Clearly, if
2204:
2200:
2196:
2191:
2170:
2150:
2147:
2144:
2139:
2135:
2131:
2128:
2108:
2088:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2041:
2021:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1975:
1971:
1946:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1888:
1885:
1882:
1877:
1873:
1860:
1857:
1849:
1848:
1832:
1829:
1826:
1823:
1812:
1798:
1795:
1792:
1788:
1767:
1762:
1758:
1754:
1751:
1725:
1722:
1719:
1714:
1710:
1706:
1703:
1700:
1695:
1692:
1689:
1684:
1680:
1656:
1645:
1644:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1592:
1589:
1586:
1582:
1561:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1509:
1498:
1495:
1492:
1488:
1483:
1477:
1474:
1471:
1466:
1462:
1458:
1453:
1450:
1447:
1442:
1438:
1434:
1429:
1426:
1423:
1418:
1414:
1409:
1405:
1402:
1398:
1392:
1389:
1386:
1381:
1377:
1373:
1368:
1365:
1362:
1357:
1353:
1349:
1344:
1341:
1338:
1333:
1329:
1324:
1320:
1316:
1292:
1288:
1282:
1279:
1276:
1271:
1267:
1263:
1258:
1255:
1252:
1247:
1243:
1239:
1234:
1231:
1228:
1223:
1219:
1214:
1191:
1188:
1185:
1180:
1176:
1172:
1167:
1164:
1161:
1156:
1152:
1130:
1124:
1121:
1118:
1113:
1109:
1105:
1100:
1097:
1094:
1089:
1085:
1081:
1076:
1073:
1070:
1065:
1061:
1056:
1035:
1031:
1025:
1022:
1019:
1014:
1010:
1006:
1001:
998:
995:
990:
986:
982:
977:
974:
971:
966:
962:
957:
936:
933:
930:
927:
924:
921:
918:
915:
893:
890:
887:
882:
878:
874:
871:
868:
863:
860:
857:
852:
848:
844:
841:
836:
832:
828:
825:
813:
810:
797:
794:
791:
788:
785:
765:
762:
759:
756:
753:
733:
713:
710:
707:
704:
701:
681:
678:
675:
672:
669:
649:
629:
609:
606:
603:
576:
573:
560:
555:
551:
547:
544:
524:
504:
484:
464:
459:
456:
453:
450:
446:
442:
439:
419:
396:
374:
370:
366:
346:
326:
306:
284:
281:
278:
275:
271:
267:
247:
227:
207:
187:
167:
147:
127:
124:
121:
118:
115:
112:
109:
106:
103:
100:
97:
94:
74:
71:
68:
65:
62:
59:
56:
44:
41:
9:
6:
4:
3:
2:
5236:
5225:
5222:
5220:
5217:
5216:
5214:
5200:
5196:
5191:
5186:
5183:(1): 64–119.
5182:
5178:
5177:
5172:
5165:
5157:
5153:
5149:
5145:
5140:
5135:
5131:
5127:
5120:
5112:
5108:
5104:
5100:
5096:
5092:
5085:
5076:
5071:
5064:
5056:
5052:
5048:
5044:
5040:
5036:
5029:
5021:
5017:
5013:
5009:
5005:
5001:
4994:
4986:
4982:
4978:
4974:
4970:
4966:
4959:
4951:
4947:
4943:
4939:
4935:
4931:
4924:
4916:
4912:
4908:
4904:
4900:
4896:
4889:
4881:
4877:
4873:
4869:
4865:
4861:
4860:Combinatorica
4854:
4846:
4842:
4837:
4832:
4828:
4824:
4823:
4818:
4811:
4803:
4799:
4795:
4791:
4786:
4781:
4777:
4773:
4769:
4762:
4754:
4750:
4745:
4740:
4736:
4732:
4728:
4724:
4720:
4716:
4712:
4705:
4697:
4693:
4689:
4685:
4681:
4677:
4670:
4662:
4658:
4653:
4648:
4644:
4640:
4635:
4630:
4626:
4622:
4618:
4614:
4610:
4603:
4595:
4591:
4586:
4581:
4577:
4573:
4572:
4567:
4560:
4552:
4548:
4544:
4540:
4536:
4532:
4527:
4522:
4518:
4514:
4507:
4499:
4495:
4490:
4485:
4481:
4477:
4472:
4467:
4463:
4459:
4455:
4451:
4447:
4440:
4438:
4436:
4434:
4429:
4419:
4416:
4414:
4411:
4409:
4406:
4405:
4396:
4375:
4372:
4369:
4363:
4357:
4354:
4351:
4345:
4339:
4336:
4333:
4324:
4321:
4313:
4297:
4277:
4274:
4271:
4268:
4265:
4243:
4239:
4235:
4213:
4205:
4182:
4162:
4159:
4156:
4134:
4107:
4099:
4098:
4092:
4078:
4075:
4072:
4052:
4045:depending on
4030:
4026:
3995:
3991:
3982:
3979:
3976:
3973:
3970:
3964:
3958:
3955:
3951:
3946:
3938:
3923:
3906:
3900:
3897:
3889:
3859:
3856:
3853:
3849:
3842:
3839:
3831:
3821:
3818:
3815:
3811:
3807:
3787:
3778:
3759:
3756:
3753:
3749:
3742:
3739:
3733:
3727:
3720:edges. Thus,
3707:
3703:
3696:
3690:
3670:
3650:
3630:
3610:
3590:
3570:
3545:
3542:
3539:
3535:
3528:
3503:
3499:
3492:
3489:
3481:
3478:
3475:
3471:
3464:
3461:
3455:
3449:
3444:
3441:
3419:
3399:
3377:
3373:
3367:
3364:
3361:
3357:
3353:
3350:
3345:
3341:
3320:
3295:
3283:
3277:
3274:
3264:
3260:
3256:
3251:
3247:
3234:
3220:
3217:
3212:
3208:
3202:
3198:
3175:
3171:
3167:
3164:
3161:
3156:
3152:
3131:
3111:
3089:
3085:
3079:
3075:
3071:
3068:
3063:
3059:
3055:
3050:
3047:
3044:
3040:
3019:
2997:
2993:
2972:
2969:
2966:
2963:
2960:
2940:
2937:
2932:
2928:
2921:
2918:
2915:
2907:
2904:
2901:
2897:
2893:
2888:
2884:
2861:
2857:
2853:
2850:
2847:
2842:
2838:
2817:
2797:
2774:
2771:
2768:
2745:
2725:
2705:
2702:
2699:
2694:
2690:
2683:
2680:
2677:
2669:
2666:
2663:
2659:
2633:
2621:
2615:
2612:
2602:
2598:
2594:
2589:
2585:
2561:
2558:
2555:
2552:
2549:
2524:
2520:
2494:
2472:
2468:
2464:
2461:
2458:
2453:
2449:
2445:
2440:
2436:
2415:
2395:
2372:
2369:
2366:
2343:
2334:
2315:
2311:
2285:
2243:
2223:
2198:
2194:
2168:
2148:
2145:
2142:
2137:
2133:
2129:
2126:
2106:
2083:
2080:
2077:
2074:
2071:
2065:
2062:
2053:
2039:
2019:
1996:
1990:
1987:
1981:
1973:
1969:
1961:states that,
1960:
1944:
1921:
1918:
1915:
1912:
1909:
1883:
1875:
1871:
1856:
1854:
1846:
1827:
1821:
1813:
1793:
1786:
1760:
1756:
1749:
1741:
1740:
1739:
1720:
1712:
1708:
1704:
1701:
1698:
1690:
1682:
1678:
1668:
1654:
1627:
1624:
1621:
1618:
1615:
1587:
1580:
1559:
1536:
1533:
1530:
1527:
1524:
1518:
1510:
1496:
1493:
1490:
1486:
1481:
1472:
1464:
1460:
1456:
1448:
1440:
1436:
1432:
1424:
1416:
1412:
1407:
1403:
1400:
1396:
1387:
1379:
1375:
1371:
1363:
1355:
1351:
1347:
1339:
1331:
1327:
1322:
1318:
1314:
1306:
1305:
1304:
1290:
1286:
1277:
1269:
1265:
1261:
1253:
1245:
1241:
1237:
1229:
1221:
1217:
1212:
1186:
1178:
1174:
1170:
1162:
1154:
1150:
1128:
1119:
1111:
1107:
1103:
1095:
1087:
1083:
1079:
1071:
1063:
1059:
1054:
1033:
1029:
1020:
1012:
1008:
1004:
996:
988:
984:
980:
972:
964:
960:
955:
934:
928:
925:
922:
919:
916:
888:
880:
876:
872:
869:
866:
858:
850:
846:
842:
834:
830:
823:
809:
792:
789:
786:
760:
757:
754:
731:
708:
705:
702:
679:
676:
673:
670:
667:
647:
627:
607:
604:
601:
593:
588:
586:
582:
572:
553:
549:
542:
522:
502:
482:
454:
448:
444:
437:
417:
408:
394:
372:
368:
364:
344:
324:
304:
279:
273:
269:
265:
245:
225:
205:
185:
165:
145:
125:
122:
116:
113:
110:
107:
104:
98:
95:
92:
72:
69:
66:
63:
60:
57:
54:
40:
38:
33:
31:
27:
23:
19:
5224:Graph theory
5180:
5174:
5164:
5129:
5125:
5119:
5094:
5090:
5084:
5063:
5038:
5034:
5028:
5003:
4999:
4993:
4968:
4964:
4958:
4933:
4929:
4923:
4898:
4894:
4888:
4863:
4859:
4853:
4826:
4825:. Series A.
4820:
4810:
4775:
4771:
4761:
4718:
4714:
4704:
4679:
4675:
4669:
4616:
4612:
4602:
4575:
4569:
4559:
4516:
4512:
4506:
4453:
4449:
4095:Applications
3924:
3779:
3235:
3012:is a length
2335:
2054:
1862:
1850:
1669:
1646:
1046:We say that
815:
589:
578:
571:hyperedges.
535:by removing
409:
46:
34:
17:
15:
5219:Hypergraphs
2542:. For each
2428:with parts
5213:Categories
5075:2402.17995
4778:(1): 151.
4424:References
475:copies of
22:hypergraph
5199:0021-7670
5156:1631-073X
5139:1007.1839
5111:0963-5483
5055:1042-9832
5020:1042-9832
4985:0911-0119
4950:1042-9832
4915:0097-5397
4880:0209-9683
4845:0097-3165
4794:0894-0347
4735:1042-9832
4696:1042-9832
4643:0027-8424
4594:0012-365X
4551:0003-486X
4526:0710.3032
4480:0027-8424
4269:⋅
4236:δ
4157:δ
4076:≥
3980:
3974:
3965:−
3959:
3947:≤
3857:−
3819:−
3757:−
3543:−
3479:−
3365:≠
3358:∑
3354:−
3290:∖
3278:∈
3257:∈
3199:∑
3172:α
3165:⋯
3153:α
3076:∑
3072:−
3060:α
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