592:
583:
285:
336:
722:
6342:, applies to infinite graphs. In a context where finite graphs are also being discussed it is often called the "Infinite Ramsey theorem". As intuition provided by the pictorial representation of a graph is diminished when moving from finite to infinite graphs, theorems in this area are usually phrased in
4952:
5733:
While the general bounds for the induced Ramsey numbers are exponential in the size of the graph, the behaviour is much different on special classes of graphs (in particular, sparse ones). Many of these classes have induced Ramsey numbers polynomial in the number of vertices.
2886:
maintains a list of known Ramsey graphs. Upper bounds are often considerably more difficult to establish: one either has to check all possible colourings to confirm the absence of a counterexample, or to present a mathematical argument for its absence.
4489:
2925:
A sophisticated computer program does not need to look at all colourings individually in order to eliminate all of them; nevertheless it is a very difficult computational task that existing software can only manage on small sizes. Each complete graph
4313:
5120:
5724:
For lower bounds, not much is known in general except for the fact that induced Ramsey numbers must be at least the corresponding Ramsey numbers. Some lower bounds have been obtained for some special cases (see
Special Cases).
2256:
4753:
7347:
4117:
4009:
7531:
8111:
8383:
8304:
8173:
8007:
8246:
5568:
and show that it has the desired properties with nonzero probability. The idea of using random graphs on projective planes have also previously been used in studying Ramsey properties with respect to
5537:
2491:
10241:
2363:
4683:
5264:. Roughly speaking, instead of finding a monochromatic subgraph, we are now required to find a monochromatic induced subgraph. In this variant, it is no longer sufficient to restrict our focus to
7443:
5268:, since the existence of a complete subgraph does not imply the existence of an induced subgraph. The qualitative statement of the theorem in the next section was first proven independently by
3892:
2906:
or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for
624:
A multicolour Ramsey number is a Ramsey number using 3 or more colours. There are (up to symmetries) only two non-trivial multicolour Ramsey numbers for which the exact value is known, namely
3331:, which is periodically updated. Where not cited otherwise, entries in the table below are taken from the January 2021 edition. (Note there is a trivial symmetry across the diagonal since
1607:
1479:
5250:
5185:
971:
709:, it suffices to draw an edge colouring on the complete graph on 16 vertices with 3 colours that avoids monochromatic triangles. It turns out that there are exactly two such colourings on
718:, the so-called untwisted and twisted colourings. Both colourings are shown in the figures to the right, with the untwisted colouring on the left, and the twisted colouring on the right.
1161:
641:
Suppose that we have an edge colouring of a complete graph using 3 colours, red, green and blue. Suppose further that the edge colouring has no monochromatic triangles. Select a vertex
6564:, the statement is equivalent to saying that if you split an infinite set into a finite number of sets, then one of them is infinite. This is evident. Assuming the theorem is true for
416:, are also blue then we have an entirely blue triangle. If not, then those three edges are all red and we have an entirely red triangle. Since this argument works for any colouring,
4528:
4344:
1999:
7607:
4152:
2092:
1278:
1218:
133:, that seeks regularity amid disorder: general conditions for the existence of substructures with regular properties. In this application it is a question of the existence of
4564:
8936:
1733:
4584:
1782:
8046:
7150:
1337:
1933:
1821:
7258:
7222:
7186:
7105:
7038:
1894:
1678:
8332:
144:
An extension of this theorem applies to any finite number of colours, rather than just two. More precisely, the theorem states that for any given number of colours,
6404:
4981:
1851:
6544:
6524:
6504:
6484:
6464:
6444:
6424:
6371:
4986:
4122:
was given by Erdős in 1947 and was instrumental in his introduction of the probabilistic method. There is a huge gap between these two bounds: for example, for
5452:
because the complete graph achieves a lower bound of this form (in fact, it's the same as Ramsey numbers). However, this conjecture is still open as of now.
2115:
507:, there exist precisely two such triples. Therefore, there are at most 18 non-monochromatic triangles. Therefore, at least 2 of the 20 triangles in the
4947:{\displaystyle c'_{s}{\frac {t^{\frac {s+1}{2}}}{(\log t)^{{\frac {s+1}{2}}-{\frac {1}{s-2}}}}}\leq R(s,t)\leq c_{s}{\frac {t^{s-1}}{(\log t)^{s-2}}}.}
4146:. There is no known explicit construction producing an exponential lower bound. The best known lower and upper bounds for diagonal Ramsey numbers are
2548:-coloured in the 'blurred colour'. In the former case we are finished. In the latter case, we recover our sight again and see from the definition of
7283:
8668:
562:
5665:, which remains the current best upper bound for general induced Ramsey numbers. Similar to the previous work in 2008, they showed that every
7640:
vertices – in a normal graph an edge is a set of 2 vertices. The full statement of Ramsey's theorem for hypergraphs is that for any integers
5542:
However, that was still far from the exponential bound conjectured by Erdős. It was not until 1998 when a major breakthrough was achieved by
4137:. Nevertheless, the exponential growth factors of either bound were not improved for a long time, and for the lower bound it still stands at
777:
that avoid monochromatic triangles, provided that we consider edge colourings that differ by a permutation of the colours as being the same.
8418:
657:
cannot contain any red edges, since otherwise there would be a red triangle consisting of the two endpoints of that red edge and the vertex
10020:
8507:), thus avoiding a discussion of edge colouring a graph with no edges, while others rephrase the statement of the theorem to require, in a
10704:
Bian, Zhengbing; Chudak, Fabian; Macready, William G.; Clark, Lane; Gaitan, Frank (2013), "Experimental determination of Ramsey numbers",
9445:
4020:
3921:
2799:
can be extracted from the proof of the theorem, and other arguments give lower bounds. (The first exponential lower bound was obtained by
7454:
8051:
9161:
3384:
10229:. Lecture Notes Series of the Institute for Mathematical Sciences, National University of Singapore. Vol. 28. World Scientific.
8341:
8251:
8120:
3082:-node red subgraph, and all other colourings contain a 2-node blue subgraph (that is, a pair of nodes connected with a blue edge.)
9139:
Campos, Marcelo; Griffiths, Simon; Morris, Robert; Sahasrabudhe, Julian (2023). "An exponential improvement for diagonal Ramsey".
7954:
2807:.) However, there is a vast gap between the tightest lower bounds and the tightest upper bounds. There are also very few numbers
756:
that avoid monochromatic triangles, which can be constructed by deleting any vertex from the untwisted and twisted colourings on
8186:
6911:
6142:
by iteratively applying the bound on the two-color case. The current best known bound is due to Fox and
Sudakov, which achieves
5461:
2391:
2266:
10627:
10570:
10419:
10105:
10012:
9995:
9796:
9486:
8680:
8406:
8397:, there is a significant difference in proof strength between the version of Ramsey's theorem for infinite graphs (the case
4641:
5666:
5592:
3174:
is unknown, although it is known to lie between 43 (Geoffrey Exoo (1989)) and 48 (Angeltveit and McKay (2017)), inclusive.
7365:
6196:. Furthermore, we can define the multicolor version of induced Ramsey numbers in the same way as the previous subsection.
5969:
Similar to Ramsey numbers, we can generalize the notion of induced Ramsey numbers to hypergraphs and multicolor settings.
9071:
3323:
295:
Due to the pigeonhole principle, there are at least 3 edges of the same colour (dashed purple) from an arbitrary vertex
10121:
McKay, Brendan D.; Radziszowski, Stanislaw P. (1991). "The First
Classical Ramsey Number for Hypergraphs is Computed".
3812:
8627:
10829:
10645:
10334:
4704:
1562:
1389:
452:
6266:. Using the hypergraph container method, Conlon, Dellamonica, La Fleur, Rödl and Schacht were able to show that for
5831:
3123:
graphs (that is, 2-colourings of a complete graph on 16 nodes without 4-node red or blue complete subgraphs) among
2677:
in Ramsey's theorem (and their extensions to more than two colours) are known as Ramsey numbers. The Ramsey number
884:
445:
5591:
provided an explicit construction for induced Ramsey numbers with the same bound. In fact, they showed that every
4334:
claims to have made exponential progress using an algorithmic construction relying on a graph structure called a "
10783:
9818:
9777:
Beck, József (1990). "On Size Ramsey Number of Paths, Trees and
Circuits. II". In Nešetřil, J.; Rödl, V. (eds.).
9559:. Colloquia Mathematica Societatis János Bolyai. Vol. 10. North-Holland, Amsterdam/London. pp. 323–332.
9541:. Colloquia Mathematica Societatis János Bolyai. Vol. 10. North-Holland, Amsterdam/London. pp. 585–595.
8986:
8694:
8430:
3188:
graphs, arriving at the same set of graphs through different routes. None of the 656 graphs can be extended to a
1060:
3001:. One of the best-known searching algorithms for unstructured datasets exhibits only a quadratic speedup (c.f.
2715:
will not know each other. In the language of graph theory, the Ramsey number is the minimum number of vertices,
8520:
4624:
740:, and consider the graph whose edges are precisely those edges that have the specified colour, we will get the
5937:
5190:
5125:
10776:
5640:
vertices is such that all of its edge colorings in two colors contain an induced monochromatic copy of every
4716:
4327:
3177:
In 1997, McKay, Radziszowski and Exoo employed computer-assisted graph generation methods to conjecture that
3157:
2899:
asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of
2883:
221:
8429:, and (combining Seetapun's result with others) it does not fall into one of the big five subsystems. Over
5407:
5284:
in the 1970s. Since then, there has been much research in obtaining good bounds for induced Ramsey numbers.
5277:
4484:{\displaystyle R(s,s)\leq (4-\varepsilon )^{s}{\text{ and }}R(s,t)\leq e^{-\delta t+o(s)}{\binom {s+t}{t}}.}
9963:
8455:
4308:{\displaystyle {\frac {{\sqrt {2}}s}{e}}2^{\frac {s}{2}}\leq R(s,s)\leq s^{-(c\log s)/(\log \log s)}4^{s},}
126:
125:
Ramsey's theorem is a foundational result in combinatorics. The first version of this result was proved by
10790:
project designed to find new lower bounds for various Ramsey numbers using a host of different techniques.
10202:
Neiman, David; Mackey, John; Heule, Marijn (2020-11-01). "Tighter Bounds on
Directed Ramsey Number R(7)".
7869:
nodes, contains a monochromatic complete graph on n nodes. (The directed analogue of the two possible arc
5853:
10771:
5394:
Similar to Ramsey's theorem, it is unclear a priori whether induced Ramsey numbers exist for every graph
4497:
1938:
9066:
7769:. This fact was established by Brendan McKay and Stanisław Radziszowski in 1991. Additionally, we have:
7577:
2034:
10793:
8569:
7877:
of the arcs, the analogue of "monochromatic" is "all arc-arrows point the same way"; i.e., "acyclic.")
4733:-free process" has set the best known asymptotic lower bounds for general off-diagonal Ramsey numbers,
3263:
are shown in the table below. Where the exact value is unknown, the table lists the best known bounds.
1223:
373:
1350:. The latter case is analogous. Thus the claim is true and we have completed the proof for 2 colours.
1166:
8602:
5868:
5418:) on the induced Ramsey numbers. It is interesting to ask if better bounds can be achieved. In 1974,
4533:
591:
10766:
8565:
8480:
5414:
independently proved that this is the case. However, the original proofs gave terrible bounds (e.g.
5187:
which settles a question of Erdős who offered 250 dollars for a proof that the lower limit has form
3328:
3161:
9252:
8900:
1683:
4569:
2971:
colours) if brute force is used. Therefore, the complexity for searching all possible graphs (via
1742:
618:, up to isomorphism and permutation of colors: the untwisted (left) and twisted (right) colorings.
9860:
9682:
8940:
8870:
8834:
8712:
8414:
8024:
7117:
1310:
495:(three are the same colour, two are the other colour) such triples. Therefore, there are at most
9629:
9593:
Recent advances in graph theory (Proceedings of the Second
Czechoslovak Symposium, Prague, 1974)
1899:
1787:
10824:
10549:
9959:
9247:
8470:
8465:
8018:
7231:
7195:
7159:
7078:
7011:
6931:
1863:
1640:
791:
10288:
8672:
8662:
360:
Suppose the edges of a complete graph on 6 vertices are coloured red and blue. Pick a vertex,
10787:
10430:
8317:
8176:
7825:
such that any complete graph with singly directed arcs (also called a "tournament") and with
5718:
5573:
3002:
1637:
are the vertices incident to vertex 1 in the blue and red subgraphs, respectively. Then both
582:
34:
10723:
10526:
10461:
10383:
10081:
9781:. Algorithms and Combinatorics. Vol. 5. Springer, Berlin, Heidelberg. pp. 34–45.
9473:, Algorithms and Combinatorics, vol. 5, Berlin, Heidelberg: Springer, pp. 12–28,
9373:
9090:
8784:
8731:
8114:
6318:. In particular, this result mirrors the best known bound for the usual Ramsey number when
5742:
5583:
Kohayakawa, Prömel and Rödl's bound remained the best general bound for a decade. In 2008,
5543:
2804:
2629:
is finite. The right hand side of the inequality in Lemma 2 expresses a Ramsey number for
1736:
369:
9608:"On some problems in graph theory, combinatorial analysis and combinatorial number theory"
6376:
5115:{\displaystyle c'_{s}t^{\frac {5}{2}}(\log t)^{-2}\leq R(4,t)\leq c_{s}t^{3}(\log t)^{-2}}
8:
8394:
7617:
5783:
5750:
5746:
4960:
4632:
4335:
4331:
1826:
289:
10727:
10387:
9377:
9094:
8788:
8735:
8438:
8434:
5977:
We can also generalize the induced Ramsey's theorem to a multicolor setting. For graphs
4715:, and the implicit constant was improved independently by Fiz Pontiveros, Griffiths and
10747:
10713:
10465:
10439:
10399:
10373:
10269:
10203:
10069:
9971:
9921:
9869:
9747:
9691:
9416:
9363:
9322:
9140:
9119:
9080:
9038:
8949:
8800:
8774:
8747:
8721:
8541:
7946:
6529:
6509:
6489:
6469:
6449:
6429:
6409:
6356:
5941:
3010:
2972:
10541:
10338:
4708:
10739:
10695:
10641:
10623:
10566:
10415:
10355:
10308:
10261:
10224:
10162:
10101:
9991:
9955:
9792:
9482:
9389:
9204:
9199:
9182:
8804:
8676:
8460:
5411:
5321:
5281:
3086:
1556:
1011:
10494:
10751:
10735:
10731:
10690:
10666:
10589:
10558:
10489:
10469:
10449:
10403:
10391:
10350:
10300:
10253:
10152:
10123:
Proceedings of the Second Annual ACM-SIAM Symposium on
Discrete Algorithms, SODA'91
10059:
10051:
9981:
9931:
9879:
9827:
9782:
9757:
9701:
9646:
9474:
9426:
9381:
9332:
9257:
9194:
9098:
8995:
8959:
8879:
8843:
8822:
8792:
8751:
8739:
8581:
5565:
5309:
5261:
2998:
2251:{\displaystyle R(n_{1},\dots ,n_{c})\leq R(n_{1},\dots ,n_{c-2},R(n_{c-1},n_{c})).}
54:
10808:
10025:
9537:; Hajnal, A.; Pósa, L. (1975). "Strong embeddings of graphs into colored graphs".
6930:
It is possible to deduce the finite Ramsey theorem from the infinite version by a
5403:
5273:
4727:, by analysing the triangle-free process. Furthermore, studying the more general "
2692:
gives the solution to the party problem, which asks the minimum number of guests,
392:, are blue. (If not, exchange red and blue in what follows.) If any of the edges,
10802:
10678:
10654:
10537:
10522:
10457:
10364:
Bohman, Tom; Keevash, Peter (2010), "The early evolution of the H-free process",
10077:
9986:
9958:; Schacht, Mathias (2017). "A note on induced Ramsey numbers". In Loebl, Martin;
9450:
9166:
8658:
8450:
8311:
8307:
5569:
3902:
3006:
10562:
10453:
10330:
9787:
9478:
9430:
5703:
vertices in any edge coloring in two colors. Currently, Erdős's conjecture that
4700:
676:
can contain at most 5 vertices. Similarly, the green and blue neighbourhoods of
10670:
8796:
8422:
8335:
7616:
If a suitable topological viewpoint is taken, this argument becomes a standard
6695:
is coloured the same colour in the induced colouring. Thus there is an element
5265:
73:
42:
38:
10507:
10395:
10304:
10064:
9936:
9907:
9884:
9855:
9762:
9733:
9706:
9677:
9607:
9467:"Problems and Results on Graphs and Hypergraphs: Similarities and Differences"
9466:
9407:
Mattheus, Sam; Verstraete, Jacques (5 Mar 2024). "The asymptotics of r(4,t)".
9385:
9016:
8707:
5328:
and its edges are monochromatic). The smallest possible number of vertices of
768:
It is also known that there are exactly 115 edge colourings with 3 colours on
284:
10818:
10601:
10312:
10265:
10166:
10157:
10140:
9912:
9903:
9738:
9729:
9673:
9637:
9393:
9208:
8644:
8531:). In this form, the consideration of graphs with one vertex is more natural.
8405:≥ 3). The multigraph version of the theorem is equivalent in strength to the
7620:
showing that the infinite version of the theorem implies the finite version.
6915:
6132:
5919:
5588:
5415:
5352:
Sometimes, we also consider the asymmetric version of the problem. We define
4724:
4712:
3060:
nodes with all edges coloured red serves as a counterexample and proves that
741:
725:
335:
130:
22:
10533:
10503:
10477:
9588:
9534:
9351:
7803:
7448:
It follows that the intersection of all of these sets is non-empty, and let
7342:{\displaystyle C_{k}\supseteq C_{k}^{1}\supseteq C_{k}^{2}\supseteq \cdots }
7226:
as the set of all such restrictions, a non-empty set. Continuing so, define
6910:
A stronger but unbalanced infinite form of Ramsey's theorem for graphs, the
5419:
5399:
5269:
5260:
There is a less well-known yet interesting analogue of Ramsey's theorem for
3898:
2896:
2800:
10743:
10605:
10593:
9951:
9847:
9832:
9813:
9721:
9261:
9000:
8978:
8976:
8883:
8847:
8508:
5648:
4323:
4319:
2918:
30:
9274:
8743:
10181:
8765:
Wang, Hefeng (2016). "Determining Ramsey numbers on a quantum computer".
8475:
6919:
5630:
3132:
747:
It is known that there are exactly two edge colourings with 3 colours on
10515:
A Magyar Tudományos Akadémia, Matematikai Kutató Intézetének Közleményei
6934:. Suppose the finite Ramsey theorem is false. Then there exist integers
5852:). Note that this is in contrast to the usual Ramsey numbers, where the
5564:. Their approach was to consider a suitable random graph constructed on
4566:
where it is believed these parameters could be optimized, in particular
3317:
The standard survey on the development of Ramsey number research is the
845:
exists by finding an explicit bound for it. By the inductive hypothesis
731:
If we select any colour of either the untwisted or twisted colouring on
10444:
10428:
Conlon, David (2009), "A new upper bound for diagonal Ramsey numbers",
10273:
10073:
9908:"Density theorems for bipartite graphs and related Ramsey-type results"
9651:
7629:
6343:
5936:
were obtained since then. In 2013, Conlon, Fox and Zhao showed using a
4720:
4623:; this may be stated equivalently as saying that the smallest possible
665:
has edges coloured with only two colours, namely green and blue. Since
455:. It goes as follows: Count the number of ordered triples of vertices,
9336:
8963:
8499:
Some authors restrict the values to be greater than one, for example (
7951:
In terms of the partition calculus, Ramsey's theorem can be stated as
2633:
colours in terms of Ramsey numbers for fewer colours. Therefore, any
45:. To demonstrate the theorem for two colours (say, blue and red), let
9899:
9851:
9725:
9669:
8954:
5584:
5448:. If this conjecture is true, it would be optimal up to the constant
2913:. In that case, he believes, we should attempt to destroy the aliens.
57:. Ramsey's theorem states that there exists a least positive integer
10257:
10055:
9926:
9507:
9310:
8441:
is equivalent to countable choice from finite sets in this setting.
7839:
This is the directed-graph analogue of what (above) has been called
5546:, Prömel and Rödl, who proved the first almost-exponential bound of
4112:{\displaystyle R(s,s)\geq (1+o(1)){\frac {s}{{\sqrt {2}}e}}2^{s/2},}
4004:{\displaystyle R(s,s)\leq (1+o(1)){\frac {4^{s-1}}{\sqrt {\pi s}}}.}
680:
can contain at most 5 vertices each. Since every vertex, except for
327:. But if not, each must be oppositely coloured, completing triangle
10208:
9976:
9421:
9145:
9124:
9085:
9043:
8779:
8726:
7526:{\displaystyle D_{k}=C_{k}\cap C_{k}^{1}\cap C_{k}^{2}\cap \cdots }
323:(solid black) had this colour, it would complete the triangle with
10798:
dynamic survey of small Ramsey numbers (by Stanisław
Radziszowski)
10718:
10378:
9874:
9814:"On induced Ramsey numbers for graphs with bounded maximum degree"
9752:
9696:
9446:"Mathematicians Discover Novel Way to Predict Structure in Graphs"
9368:
9327:
9103:
1006:
vertices whose edges are coloured with two colours. Pick a vertex
645:. Consider the set of vertices that have a red edge to the vertex
483:, is blue. Firstly, any given vertex will be the middle of either
10508:"On the representation of directed graphs as unions of orderings"
9950:
9138:
8106:{\displaystyle 2^{\aleph _{0}}\nrightarrow (\aleph _{1})_{2}^{2}}
8021:
showed that the Ramsey theorem does not extend to graphs of size
7762:
we know the exact value of one non-trivial Ramsey number, namely
8586:
7745:, the 'hyper-ness' of the graph. The base case for the proof is
6747:
have the same colour. By the same argument, there is an element
6669:). By the induction hypothesis, there exists an infinite subset
5911:. This result was first proven by Łuczak and Rödl in 1996, with
1383:
are both even, the induction inequality can be strengthened to:
721:
376:
we can assume at least 3 of these edges, connecting the vertex,
8896:
1291:
then so does the original graph and we are finished. Otherwise
661:. Thus, the induced edge colouring on the red neighbourhood of
138:
10100:(4 ed.). Heidelberg: Springer-Verlag. pp. 209–2010.
10042:
Dushnik, Ben; Miller, E. W. (1941). "Partially ordered sets".
9591:(1975). "Problems and results on finite and infinite graphs".
9465:
Erdös, Paul (1990), Nešetřil, Jaroslav; Rödl, Vojtěch (eds.),
8378:{\displaystyle \kappa \rightarrow (\kappa )_{2}^{<\omega }}
8310:
has strengthened this result further. On the positive side, a
5961:, where the exponent is best possible up to constant factors.
2852:
usually requires exhibiting a blue/red colouring of the graph
2654:
is finite for any number of colours. This proves the theorem.
1010:
from the graph, and partition the remaining vertices into two
684:
itself, is in one of the red, green or blue neighbourhoods of
9181:
Ajtai, Miklós; Komlós, János; Szemerédi, Endre (1980-11-01).
8385:. The existence of Ramsey cardinals cannot be proved in ZFC.
8299:{\displaystyle \aleph _{1}\nrightarrow (\aleph _{1})_{2}^{2}}
8168:{\displaystyle \aleph _{1}\nrightarrow (\aleph _{1})_{2}^{2}}
5455:
In 1984, Erdős and Hajnal claimed that they proved the bound
8002:{\displaystyle \aleph _{0}\rightarrow (\aleph _{0})_{k}^{n}}
6783:
with the same properties. Inductively, we obtain a sequence
3078:
nodes, the colouring with all edges coloured red contains a
2746:. Ramsey's theorem states that such a number exists for all
1860:
is odd, the first inequality can be strengthened, so either
565:
in 1953, as well as in the
Hungarian Math Olympiad in 1947.
9555:
Deuber, W. (1975). "A generalization of Ramsey's theorem".
3379:
499:
such triples. Secondly, for any non-monochromatic triangle
9118:
Exoo, Geoffrey (26 Oct 2023). "A Lower Bound for R(5,6)".
8821:
McKay, Brendan D.; Radziszowski, Stanislaw P. (May 1995).
8241:{\displaystyle \aleph _{1}\nrightarrow _{\aleph _{1}}^{2}}
6466:
different colours. Then there exists some infinite subset
6184:
We can extend the definition of induced Ramsey numbers to
5378:
using only red or blue contains a red induced subgraph of
5370:
to be the smallest possible number of vertices of a graph
3235:
have not been improved since 1965 and 1972, respectively.
8180:
7574:, and continuing doing so, one constructs a colouring of
5699:
contains an induced monochromatic copy of every graph on
5532:{\displaystyle r_{\text{ind}}(H)\leq 2^{2^{k^{1+o(1)}}}.}
2486:{\displaystyle R(n_{1},\dots ,n_{c-2},R(n_{c-1},n_{c})),}
568:
10242:"Ramsey's theorem in the hierarchy of choice principles"
10141:"A lower bound on the hypergraph Ramsey number R(4,5;3)"
10096:
Diestel, Reinhard (2010). "Chapter 8, Infinite Graphs".
9575:
The dimension of a graph and generalized Ramsey theorems
8437:, whereas the converse implication does not hold, since
2997:
The situation is unlikely to improve with the advent of
2358:{\displaystyle R(n_{1},\dots ,n_{c-2},R(n_{c-1},n_{c}))}
491:(four are the same colour, one is the other colour), or
10703:
10600:
9037:
Angeltveit, Vigleik (31 Dec 2023). "R(3,10) <= 41".
5617:
contains an induced monochromatic copy of any graph on
5122:, but a 2023 preprint has improved the lower bound to
3127:
different 2-colourings of 16-node graphs, and only one
8388:
8338:
axiomatically defined to satisfy the related formula:
6925:
4678:{\displaystyle \Theta \left({\sqrt {n\log n}}\right).}
4326:, respectively; a 2023 preprint by Campos, Griffiths,
1566:
185:, such that if the edges of a complete graph of order
8903:
8897:
Vigleik
Angeltveit; Brendan McKay (September 2018). "
8425:, the graph version of the theorem is weaker than ACA
8421:
in reverse mathematics. In contrast, by a theorem of
8344:
8320:
8254:
8189:
8123:
8054:
8027:
7957:
7858:
such that any 2-colouring of the edges of a complete
7832:
nodes contains an acyclic (also called "transitive")
7580:
7457:
7368:
7286:
7234:
7198:
7162:
7120:
7081:
7014:
6914:, states that every infinite graph contains either a
6532:
6512:
6492:
6472:
6452:
6432:
6412:
6379:
6359:
5717:
remains open and is one of the important problems in
5464:
5193:
5128:
4989:
4963:
4756:
4644:
4572:
4536:
4500:
4347:
4155:
4023:
3924:
3815:
2394:
2269:
2118:
2037:
1941:
1902:
1866:
1829:
1790:
1745:
1686:
1643:
1565:
1392:
1313:
1226:
1169:
1063:
887:
129:. This initiated the combinatorial theory now called
10329:
10183:
Partial Answer to Puzzle #27: A Ramsey-like quantity
9627:
9311:"Dynamic concentration of the triangle-free process"
9180:
8977:
Brendan D. McKay, Stanisław P. Radziszowski (1997).
7438:{\displaystyle |C_{k}|\leq c^{\frac {k!}{n!(k-n)!}}}
6969:-colourings of without a monochromatic set of size
5651:, Fox and Sudakov were able to improve the bound to
299:. Calling 3 of the vertices terminating these edges
8433:, however, the graph version implies the classical
6992:to (by ignoring the colour of all sets containing
6954:-colouring of without a monochromatic set of size
6918:independent set, or an infinite clique of the same
6891:such that this colour will be the same. Take these
3221:, only weak bounds are available. Lower bounds for
790:The theorem for the 2-colour case can be proved by
544:. The unique colouring is shown to the right. Thus
10134:
10132:
9406:
9067:"New Lower Bounds for 28 Classical Ramsey Numbers"
9014:
8930:
8377:
8326:
8298:
8240:
8167:
8105:
8040:
8001:
7601:
7525:
7437:
7341:
7252:
7216:
7180:
7144:
7099:
7032:
6538:
6518:
6498:
6478:
6458:
6438:
6418:
6398:
6365:
5531:
5244:
5179:
5114:
4975:
4946:
4677:
4578:
4558:
4522:
4483:
4307:
4111:
4003:
3886:
3362:Values / known bounding ranges for Ramsey numbers
2734:, such that all undirected simple graphs of order
2485:
2357:
2250:
2086:
1993:
1927:
1888:
1845:
1815:
1776:
1727:
1672:
1601:
1473:
1331:
1272:
1212:
1155:
965:
10805:(contains lower and upper bounds up to R(19, 19))
10201:
10120:
9275:"The Triangle-Free Process and the Ramsey Number
8979:"Subgraph Counting Identities and Ramsey Numbers"
8820:
7798:It is also possible to define Ramsey numbers for
7741:. This theorem is usually proved by induction on
7636:-hypergraph is a graph whose "edges" are sets of
6188:-uniform hypergraphs by simply changing the word
6043:colors contain an induced subgraph isomorphic to
4472:
4451:
3887:{\displaystyle R(r,s)\leq {\binom {r+s-2}{r-1}}.}
3875:
3840:
487:(all edges from the vertex are the same colour),
10816:
10681:(1975), "Ramsey's theorem – a new lower bound",
10620:Combinatorial Methods with Computer Applications
10480:(1947), "Some remarks on the theory of graphs",
9856:"Extremal results in sparse pseudorandom graphs"
9720:
8861:Exoo, Geoffrey (March 1989). "A lower bound for
7613:. This contradicts the infinite Ramsey theorem.
6373:be some infinite set and colour the elements of
6031:to be the minimum number of vertices in a graph
5625:in two colors. In particular, for some constant
2369:colours. Now 'go colour-blind' and pretend that
10414:(5th ed.), Prentice-Hall, pp. 77–82,
10129:
9954:; Dellamonica Jr., Domingos; La Fleur, Steven;
9628:Kohayakawa, Y.; Prömel, H.J.; Rödl, V. (1998).
9533:
9162:"A Very Big Small Leap Forward in Graph Theory"
8657:
7112:Similarly, the restriction of any colouring in
6880:. Further, there are infinitely many values of
3005:) relative to classical computers, so that the
1602:{\displaystyle \textstyle \sum _{i=1}^{t}d_{i}}
1474:{\displaystyle R(r,s)\leq R(r-1,s)+R(r,s-1)-1.}
804:. It is clear from the definition that for all
72:for which every blue-red edge colouring of the
33:forms, states that one will find monochromatic
10659:Proceedings of the London Mathematical Society
9030:
8695:2.6 Ramsey Theory from Mathematics Illuminated
563:William Lowell Putnam Mathematical Competition
372:) at least 3 of them must be the same colour.
10580:Exoo, G. (1989), "A lower bound for R(5,5)",
10532:
10363:
10041:
9846:
9349:
9308:
9064:
8528:
7726:, the hypergraph must contain a complete sub-
7555:. Therefore, by unrestricting a colouring in
4711:, the lower bound was obtained originally by
1555:-th vertex in the blue subgraph, then by the
966:{\displaystyle R(r,s)\leq R(r-1,s)+R(r,s-1).}
688:, the entire complete graph can have at most
10286:
10021:Mathematical Institute, University of Oxford
9812:Łuczak, Tomasz; Rödl, Vojtěch (March 1996).
8816:
8814:
8564:
8401:= 2) and for infinite multigraphs (the case
6328:
2878:subgraph. Such a counterexample is called a
2012:and the proof is complete, or it has a blue
1326:
1320:
444:. The popular version of this is called the
10287:Forster, T.E.; Truss, J.K. (January 2007).
10222:
10138:
10086:. See in particular Theorems 5.22 and 5.23.
9352:"The early evolution of the H-free process"
9060:
9058:
9056:
9054:
2890:
2380:are the same colour. Thus the graph is now
1156:{\displaystyle R(r-1,s)+R(r,s-1)=|M|+|N|+1}
830:. This starts the induction. We prove that
10502:
9898:
9811:
9668:
9443:
9350:Bohman, Tom; Keevash, Peter (2010-08-01).
9309:Bohman, Tom; Keevash, Peter (2020-11-17).
9222:Kim, Jeong Han (1995), "The Ramsey Number
9036:
7807:
7685:such that if the hyperedges of a complete
5572:and the induced Ramsey problem on bounded
5308:using two colors contains a monochromatic
3184:. They were able to construct exactly 656
649:. This is called the red neighbourhood of
114:signifies an integer that depends on both
10717:
10694:
10493:
10443:
10377:
10354:
10207:
10156:
10063:
9985:
9975:
9935:
9925:
9883:
9873:
9831:
9786:
9761:
9751:
9705:
9695:
9650:
9420:
9367:
9326:
9251:
9198:
9187:Journal of Combinatorial Theory, Series A
9144:
9123:
9102:
9084:
9065:Exoo, Geoffrey; Tatarevic, Milos (2015).
9042:
8999:
8953:
8811:
8778:
8725:
8705:
8585:
8560:
8558:
8556:
7583:
6907:'s to get the desired monochromatic set.
5422:conjectured that there exists a constant
5374:such that every coloring of the edges of
3806:may be applied inductively to prove that
1620:. Assume without loss of generality that
519:Conversely, it is possible to 2-colour a
10811:(Contains R(5, 5) > 42 counter-proof)
10657:(1930), "On a problem of formal logic",
9734:"On two problems in graph Ramsey theory"
9111:
9051:
8664:Ten Lectures on the Probabilistic Method
7940:
7352:and each set is non-empty. Furthermore,
7049:which are restrictions of colourings in
5245:{\displaystyle c'_{s}t^{3}(\log t)^{-d}}
5180:{\displaystyle c'_{s}t^{3}(\log t)^{-4}}
2707:, that must be invited so that at least
2611:. In either case the proof is complete.
2503:mono-chromatically coloured with colour
720:
334:
283:
10677:
10409:
10095:
9132:
8639:
8637:
8500:
6338:A further result, also commonly called
6035:such that any coloring of the edges of
5304:such that any coloring of the edges of
2097:
2024:which along with vertex 1 makes a blue
141:of connected edges of just one colour.
41:(with colours) of a sufficiently large
16:Statement in mathematical combinatorics
10817:
10653:
10635:
10427:
9968:A Journey Through Discrete Mathematics
9577:(Master's thesis). Charles University.
9554:
9159:
9008:
8553:
8504:
7752:, which is exactly the theorem above.
7609:without any monochromatic set of size
10617:
10542:"A combinatorial problem in geometry"
10476:
10239:
10179:
10145:Contributions to Discrete Mathematics
9664:
9662:
9605:
9595:. Academia, Prague. pp. 183–192.
9587:
9568:
9566:
9550:
9548:
9529:
9527:
9464:
8524:
8409:, making it part of the subsystem ACA
7543:is the restriction of a colouring in
6211:vertices. Define the tower function
5621:vertices in any coloring of edges of
5389:
5300:vertices. Then, there exists a graph
4600:, it is known that they are of order
3085:Using induction inequalities and the
2815:for which we know the exact value of
1540:and consider a two-coloured graph of
10579:
10341:(1980), "A note on Ramsey numbers",
10289:"Ramsey's theorem and König's Lemma"
9970:. Springer, Cham. pp. 357–366.
9776:
9572:
9444:Cepelewicz, Jordana (22 June 2023).
9117:
8860:
8764:
8634:
7628:The theorem can also be extended to
6980:, the restriction of a colouring in
6113:It is possible to derive a bound on
4589:For the off-diagonal Ramsey numbers
2959:edges, so there would be a total of
2388:-coloured. Due to the definition of
10796:Electronic Journal of Combinatorics
9221:
9072:Electronic Journal of Combinatorics
8574:Electronic Journal of Combinatorics
8389:Relationship to the axiom of choice
6926:Infinite version implies the finite
6050:where all edges are colored in the
4523:{\displaystyle \varepsilon =2^{-7}}
3897:In particular, this result, due to
3324:Electronic Journal of Combinatorics
2365:vertices and colour its edges with
1994:{\displaystyle |M|\geq p=R(r-1,s).}
528:without creating any monochromatic
91:vertices contains a blue clique on
13:
10640:, Addison-Wesley, pp. 16–17,
10139:Dybizbański, Janusz (2018-12-31).
9659:
9563:
9545:
9524:
8603:"Party problems and Ramsey theory"
8600:
8272:
8256:
8222:
8207:
8191:
8141:
8125:
8079:
8061:
8029:
7975:
7959:
7793:
7602:{\displaystyle \mathbb {N} ^{(n)}}
6333:
5964:
4645:
4455:
3844:
2087:{\displaystyle |N|\geq q=R(r,s-1)}
14:
10841:
10759:
10240:Blass, Andreas (September 1977).
10010:
8708:"Quantum algorithms: an overview"
8248:, a much stronger statement than
7802:graphs; these were introduced by
7718:different colours, then for some
5255:
3074:; among colourings of a graph on
2742:, or an independent set of order
2711:will know each other or at least
2657:
1273:{\displaystyle |N|\geq R(r,s-1).}
1163:vertices, it follows that either
212:different colours, then for some
9557:Infinite and Finite Sets, Vol. 1
9539:Infinite and Finite Sets, Vol. 1
9315:Random Structures and Algorithms
9240:Random Structures and Algorithms
8407:arithmetical comprehension axiom
7737:whose hyperedges are all colour
7623:
6172:is a constant depending only on
5728:
4338:", improving the upper bound to
1213:{\displaystyle |M|\geq R(r-1,s)}
785:
590:
581:
446:theorem on friends and strangers
364:. There are 5 edges incident to
10612:, New York: John Wiley and Sons
10495:10.1090/S0002-9904-1947-08785-1
10280:
10233:
10216:
10195:
10180:Smith, Warren D.; Exoo, Geoff,
10173:
10114:
10089:
10044:American Journal of Mathematics
10035:
10004:
9944:
9892:
9840:
9819:Journal of Combinatorial Theory
9805:
9770:
9714:
9621:
9599:
9581:
9500:
9458:
9437:
9400:
9343:
9302:
9267:
9215:
9174:
9153:
8987:Journal of Combinatorial Theory
8970:
8890:
8854:
8758:
6557:, the size of the subsets. For
6553:: The proof is by induction on
4559:{\displaystyle \delta =50^{-1}}
3016:
2493:such a graph contains either a
10803:Ramsey Number – from MathWorld
10736:10.1103/PhysRevLett.111.130505
10293:Archive for Mathematical Logic
10223:Hirschfeldt, Denis R. (2014).
9678:"Induced Ramsey-type theorems"
9615:Graph Theory and Combinatorics
8919:
8907:
8699:
8688:
8651:
8620:
8594:
8534:
8493:
8358:
8351:
8348:
8282:
8268:
8217:
8203:
8151:
8137:
8089:
8075:
7985:
7971:
7968:
7594:
7588:
7425:
7413:
7385:
7370:
6387:
6380:
6179:
5972:
5517:
5511:
5481:
5475:
5230:
5217:
5165:
5152:
5100:
5087:
5061:
5049:
5031:
5018:
4923:
4910:
4875:
4863:
4809:
4796:
4443:
4437:
4411:
4399:
4382:
4369:
4363:
4351:
4287:
4269:
4261:
4246:
4232:
4220:
4177:
4174:
4168:
4156:
4066:
4063:
4057:
4045:
4039:
4027:
3967:
3964:
3958:
3946:
3940:
3928:
3831:
3819:
3762:
2965:graphs to search through (for
2477:
2474:
2442:
2398:
2352:
2349:
2317:
2273:
2242:
2239:
2207:
2163:
2154:
2122:
2081:
2063:
2047:
2039:
1985:
1967:
1951:
1943:
1912:
1904:
1876:
1868:
1839:
1831:
1800:
1792:
1755:
1747:
1696:
1688:
1653:
1645:
1613:is odd, there must be an even
1462:
1444:
1435:
1417:
1408:
1396:
1264:
1246:
1236:
1228:
1207:
1189:
1179:
1171:
1143:
1135:
1127:
1119:
1112:
1094:
1085:
1067:
1057:is red. Because the graph has
957:
939:
930:
912:
903:
891:
451:An alternative proof works by
276:
1:
10809:Ramsey Number – Geoffrey Exoo
10323:
10246:The Journal of Symbolic Logic
8931:{\displaystyle R(5,5)\leq 48}
6873:depends only on the value of
6808:such that the colour of each
6164:is the number of vertices of
5929:. More reasonable bounds for
5867:is linear (since trees are 1-
5316:(i.e. an induced subgraph of
3139:colourings. (It follows that
2757:By symmetry, it is true that
2263:Consider a complete graph of
1728:{\displaystyle |N|=t-1-d_{1}}
1021:, such that for every vertex
979:Consider a complete graph on
600:The only two 3-colourings of
235:. The special case above has
220:, it must contain a complete
10696:10.1016/0097-3165(75)90071-0
10410:Brualdi, Richard A. (2010),
10356:10.1016/0097-3165(80)90030-8
9987:10.1007/978-3-319-44479-6_13
9779:Mathematics of Ramsey Theory
9471:Mathematics of Ramsey Theory
9200:10.1016/0097-3165(80)90030-8
9160:Sloman, Leila (2 May 2023).
7806: and L. Moser (
6946:such that for every integer
6912:Erdős–Dushnik–Miller theorem
5382:or blue induced subgraph of
5287:
4579:{\displaystyle \varepsilon }
4014:An exponential lower bound,
3035:, and, more generally, that
2738:, contain a clique of order
1777:{\displaystyle |M|\geq p-1,}
95:vertices or a red clique on
7:
10772:Encyclopedia of Mathematics
10618:Gross, Jonathan L. (2008),
10563:10.1007/978-0-8176-4842-8_3
10454:10.4007/annals.2009.170.941
9788:10.1007/978-3-642-72905-8_4
9479:10.1007/978-3-642-72905-8_2
9431:10.4007/annals.2024.199.2.8
8444:
8041:{\displaystyle \aleph _{1}}
7145:{\displaystyle C_{k+1}^{1}}
6657:-element subset (to get an
5856:(now proven) tells us that
5790:vertices, it is known that
5757:vertices, it is known that
3089:, it can be concluded that
3028:. It is easy to prove that
1332:{\displaystyle M\cup \{v\}}
672:, the red neighbourhood of
653:. The red neighbourhood of
561:was one of the problems of
271:
231:whose edges are all colour
10:
10846:
10622:, CRC Press, p. 458,
10412:Introductory Combinatorics
9854:; Zhao, Yufei (May 2014).
9183:"A note on Ramsey numbers"
8797:10.1103/PhysRevA.93.032301
8706:Montanaro, Ashley (2016).
7944:
5886:, it was conjectured that
1928:{\displaystyle |N|\geq q.}
1816:{\displaystyle |N|\geq q.}
1353:In this 2-colour case, if
374:Without loss of generality
10396:10.1007/s00222-010-0247-x
10305:10.1007/s00153-006-0025-z
9937:10.1007/s00493-009-2475-5
9885:10.1016/j.aim.2013.12.004
9763:10.1007/s00493-012-2710-3
9707:10.1016/j.aim.2008.07.009
9386:10.1007/s00222-010-0247-x
9230:) has order of magnitude
8610:Austr. Math. Soc. Gazette
8529:Erdős & Szekeres 1935
7253:{\displaystyle C_{k}^{m}}
7217:{\displaystyle C_{k}^{2}}
7190:, allowing one to define
7181:{\displaystyle C_{k}^{1}}
7100:{\displaystyle C_{k}^{1}}
7073:is not empty, neither is
7033:{\displaystyle C_{k}^{1}}
6546:all have the same colour.
6329:Extensions of the theorem
6207:-uniform hypergraph with
6131:which is approximately a
3156:was first established by
2614:Lemma 1 implies that any
1889:{\displaystyle |M|\geq p}
1673:{\displaystyle |M|=d_{1}}
554:The task of proving that
428:contains a monochromatic
148:, and any given integers
10830:Theorems in graph theory
10683:J. Combin. Theory Ser. A
10671:10.1112/plms/s2-30.1.264
10343:J. Combin. Theory Ser. A
10158:10.11575/cdm.v13i2.62416
9630:"Induced Ramsey Numbers"
9356:Inventiones Mathematicae
9015:Stanisław Radziszowski.
8486:
8456:Paris–Harrington theorem
7536:Then every colouring in
7042:to be the colourings in
5878:with number of vertices
5816:. It is also known that
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3013:in the number of nodes.
2891:Computational complexity
2833:Computing a lower bound
780:
692:vertices. Thus, we have
10706:Physical Review Letters
10582:Journal of Graph Theory
9861:Advances in Mathematics
9683:Advances in Mathematics
8941:Journal of Graph Theory
8871:Journal of Graph Theory
8835:Journal of Graph Theory
8713:npj Quantum Information
8566:Radziszowski, Stanisław
8415:second-order arithmetic
8327:{\displaystyle \kappa }
8179:showed that in fact in
7821:be the smallest number
6922:as the original graph.
6765:and an infinite subset
6704:and an infinite subset
2990:colourings and at most
1280:In the former case, if
569:A multicolour example:
475:, is red and the edge,
10636:Harary, Frank (1972),
10594:10.1002/jgt.3190130113
10550:Compositio Mathematica
10482:Bull. Amer. Math. Soc.
9833:10.1006/jctb.1996.0025
9262:10.1002/rsa.3240070302
9001:10.1006/jctb.1996.1741
8932:
8884:10.1002/jgt.3190130113
8848:10.1002/jgt.3190190304
8570:"Small Ramsey Numbers"
8471:Van der Waerden number
8466:Infinite Ramsey theory
8379:
8328:
8300:
8242:
8169:
8107:
8042:
8003:
7854:, the smallest number
7664:, there is an integer
7603:
7527:
7439:
7343:
7254:
7218:
7182:
7146:
7101:
7034:
6932:proof by contradiction
6540:
6520:
6500:
6480:
6460:
6440:
6420:
6400:
6367:
5533:
5426:such that every graph
5398:. In the early 1970s,
5246:
5181:
5116:
4977:
4948:
4679:
4580:
4560:
4524:
4485:
4309:
4113:
4005:
3888:
3329:Stanisław Radziszowski
3162:Stanisław Radziszowski
2923:
2572:we must have either a
2487:
2359:
2252:
2094:is treated similarly.
2088:
1995:
1929:
1890:
1847:
1817:
1778:
1729:
1674:
1603:
1587:
1475:
1333:
1274:
1214:
1157:
967:
728:
609:with no monochromatic
467:, such that the edge,
357:
348:with no monochromatic
332:
10788:distributed computing
10431:Annals of Mathematics
9512:www.erdosproblems.com
9409:Annals of Mathematics
8933:
8744:10.1038/npjqi.2015.23
8628:"Party Acquaintances"
8380:
8329:
8301:
8243:
8170:
8113:. In particular, the
8108:
8043:
8004:
7941:Uncountable cardinals
7836:-node subtournament.
7730:-hypergraph of order
7689:-hypergraph of order
7604:
7528:
7440:
7344:
7273:Now, for any integer
7255:
7219:
7183:
7147:
7102:
7035:
6541:
6521:
6501:
6481:
6461:
6441:
6421:
6401:
6368:
5854:Burr–Erdős conjecture
5719:extremal graph theory
5534:
5334:induced Ramsey number
5247:
5182:
5117:
4978:
4949:
4680:
4581:
4561:
4525:
4486:
4310:
4114:
4006:
3889:
3119:. There are only two
2894:
2784:. An upper bound for
2488:
2360:
2253:
2089:
1996:
1930:
1891:
1848:
1818:
1779:
1730:
1675:
1604:
1567:
1551:is the degree of the
1476:
1346:by the definition of
1334:
1275:
1215:
1158:
968:
724:
339:A 2-edge-labeling of
338:
287:
164:, there is a number,
135:monochromatic subsets
10506:; Moser, L. (1964),
9606:Erdős, Paul (1984).
8901:
8342:
8318:
8252:
8187:
8121:
8115:continuum hypothesis
8052:
8025:
7955:
7862:directed graph with
7618:compactness argument
7578:
7455:
7366:
7284:
7232:
7196:
7160:
7118:
7079:
7012:
6999:) is a colouring in
6721:-element subsets of
6640:-element subsets of
6597:-element subsets of
6530:
6510:
6490:
6470:
6450:
6430:
6410:
6399:{\displaystyle ^{n}}
6377:
6357:
6192:in the statement to
5903:, for some constant
5462:
5191:
5126:
4987:
4961:
4754:
4688:The upper bound for
4642:
4570:
4534:
4498:
4345:
4153:
4021:
3922:
3905:, implies that when
3813:
3021:As described above,
2871:subgraph and no red
2805:probabilistic method
2392:
2267:
2116:
2098:Case of more colours
2035:
1939:
1900:
1864:
1827:
1788:
1743:
1737:Pigeonhole principle
1684:
1641:
1609:is even. Given that
1563:
1390:
1311:
1224:
1167:
1061:
885:
370:pigeonhole principle
10728:2013PhRvL.111m0505B
10388:2010InMat.181..291B
9378:2010InMat.181..291B
9095:2015arXiv150402403E
8789:2016PhRvA..93c2301W
8754:– via Nature.
8736:2016npjQI...215023M
8601:Do, Norman (2006).
8572:. Dynamic Surveys.
8419:big five subsystems
8395:reverse mathematics
8374:
8295:
8237:
8164:
8102:
7998:
7648:, and any integers
7516:
7498:
7332:
7314:
7249:
7213:
7177:
7141:
7096:
7029:
6686:-element subset of
6665:-element subset of
6506:such that the size
6054:-th color for some
5942:pseudorandom graphs
5882:and bounded degree
5434:vertices satisfies
5206:
5141:
5002:
4983:the bounds become
4976:{\displaystyle s=4}
4769:
4633:triangle-free graph
4625:independence number
3388:
3167:The exact value of
3135:of order 17) among
2005:subgraph has a red
1846:{\displaystyle |M|}
1525:are both even. Let
516:are monochromatic.
290:proof without words
10604:; Rothschild, B.;
10065:10338.dmlcz/100377
9960:Nešetřil, Jaroslav
9652:10.1007/PL00009828
8928:
8481:Erdős–Rado theorem
8375:
8357:
8324:
8296:
8281:
8238:
8216:
8165:
8150:
8103:
8088:
8038:
7999:
7984:
7947:Partition calculus
7714:are coloured with
7599:
7562:to a colouring in
7523:
7502:
7484:
7435:
7339:
7318:
7300:
7250:
7235:
7214:
7199:
7178:
7163:
7142:
7121:
7097:
7082:
7030:
7015:
6916:countably infinite
6713:such that all the
6636:-colouring of the
6589:-colouring of the
6574:, we prove it for
6536:
6516:
6496:
6476:
6456:
6436:
6416:
6396:
6363:
6310:depending on only
6306:for some constant
5907:depending only on
5560:for some constant
5529:
5390:History and bounds
5242:
5194:
5177:
5129:
5112:
4990:
4973:
4944:
4757:
4675:
4576:
4556:
4520:
4481:
4305:
4109:
4001:
3884:
3361:
3310:for all values of
3003:Grover's algorithm
2483:
2355:
2248:
2084:
1991:
1925:
1886:
1843:
1813:
1774:
1725:
1670:
1629:is even, and that
1599:
1598:
1471:
1329:
1270:
1210:
1153:
963:
729:
690:1 + 5 + 5 + 5 = 16
358:
333:
208:are coloured with
10629:978-1-58488-743-0
10572:978-0-8176-4841-1
10421:978-0-13-602040-0
10226:Slicing the Truth
10107:978-3-662-53621-6
9997:978-3-319-44478-9
9798:978-3-642-72907-2
9573:Rödl, V. (1973).
9488:978-3-642-72905-8
9337:10.1002/rsa.20973
9290:bookstore.ams.org
8964:10.1002/jgt.22235
8767:Physical Review A
8682:978-0-89871-325-1
8461:Sim (pencil game)
8019:Wacław Sierpiński
7432:
7262:for all integers
6950:, there exists a
6644:, by just adding
6632:We then induce a
6610:be an element of
6539:{\displaystyle M}
6519:{\displaystyle n}
6499:{\displaystyle X}
6479:{\displaystyle M}
6459:{\displaystyle c}
6439:{\displaystyle n}
6419:{\displaystyle X}
6366:{\displaystyle X}
5687:and edge density
5566:projective planes
5472:
5262:induced subgraphs
5015:
4939:
4855:
4850:
4829:
4792:
4666:
4470:
4394:
4211:
4197:
4188:
4135:(10, 10) ≤ 48,620
4086:
4080:
3996:
3995:
3873:
3760:
3759:
3087:handshaking lemma
2999:quantum computers
1735:are even. By the
1557:Handshaking lemma
55:positive integers
10837:
10780:
10767:"Ramsey theorem"
10754:
10721:
10699:
10698:
10673:
10650:
10632:
10613:
10596:
10575:
10546:
10538:Szekeres, George
10529:
10512:
10498:
10497:
10472:
10447:
10424:
10406:
10381:
10359:
10358:
10339:Szemerédi, Endre
10317:
10316:
10284:
10278:
10277:
10237:
10231:
10230:
10220:
10214:
10213:
10211:
10199:
10193:
10192:
10191:
10190:
10177:
10171:
10170:
10160:
10136:
10127:
10126:
10118:
10112:
10111:
10093:
10087:
10085:
10067:
10039:
10033:
10032:
10030:
10024:. Archived from
10017:
10008:
10002:
10001:
9989:
9979:
9948:
9942:
9941:
9939:
9929:
9896:
9890:
9889:
9887:
9877:
9844:
9838:
9837:
9835:
9809:
9803:
9802:
9790:
9774:
9768:
9767:
9765:
9755:
9718:
9712:
9711:
9709:
9699:
9690:(6): 1771–1800.
9666:
9657:
9656:
9654:
9634:
9625:
9619:
9618:
9612:
9603:
9597:
9596:
9585:
9579:
9578:
9570:
9561:
9560:
9552:
9543:
9542:
9531:
9522:
9521:
9519:
9518:
9508:"Erdős Problems"
9504:
9498:
9497:
9496:
9495:
9462:
9456:
9455:
9441:
9435:
9434:
9424:
9404:
9398:
9397:
9371:
9347:
9341:
9340:
9330:
9306:
9300:
9299:
9297:
9296:
9285:
9271:
9265:
9264:
9255:
9219:
9213:
9212:
9202:
9178:
9172:
9171:
9157:
9151:
9150:
9148:
9136:
9130:
9129:
9127:
9115:
9109:
9108:
9106:
9088:
9062:
9049:
9048:
9046:
9034:
9028:
9027:
9025:
9023:
9012:
9006:
9005:
9003:
8983:
8974:
8968:
8967:
8957:
8937:
8935:
8934:
8929:
8894:
8888:
8887:
8867:
8858:
8852:
8851:
8831:
8818:
8809:
8808:
8782:
8762:
8756:
8755:
8729:
8703:
8697:
8692:
8686:
8685:
8655:
8649:
8648:
8641:
8632:
8631:
8624:
8618:
8617:
8607:
8598:
8592:
8591:
8589:
8562:
8545:
8538:
8532:
8518:
8514:
8497:
8384:
8382:
8381:
8376:
8373:
8365:
8333:
8331:
8330:
8325:
8305:
8303:
8302:
8297:
8294:
8289:
8280:
8279:
8264:
8263:
8247:
8245:
8244:
8239:
8236:
8231:
8230:
8229:
8215:
8214:
8199:
8198:
8177:Stevo Todorčević
8174:
8172:
8171:
8166:
8163:
8158:
8149:
8148:
8133:
8132:
8112:
8110:
8109:
8104:
8101:
8096:
8087:
8086:
8071:
8070:
8069:
8068:
8048:by showing that
8047:
8045:
8044:
8039:
8037:
8036:
8008:
8006:
8005:
8000:
7997:
7992:
7983:
7982:
7967:
7966:
7936:
7928:
7921:
7914:
7907:
7900:
7893:
7886:
7868:
7857:
7853:
7835:
7831:
7824:
7820:
7789:
7782:
7775:
7768:
7761:
7751:
7744:
7740:
7736:
7729:
7725:
7721:
7717:
7713:
7688:
7684:
7663:
7647:
7643:
7639:
7635:
7612:
7608:
7606:
7605:
7600:
7598:
7597:
7586:
7573:
7561:
7554:
7542:
7532:
7530:
7529:
7524:
7515:
7510:
7497:
7492:
7480:
7479:
7467:
7466:
7444:
7442:
7441:
7436:
7434:
7433:
7431:
7405:
7397:
7388:
7383:
7382:
7373:
7358:
7348:
7346:
7345:
7340:
7331:
7326:
7313:
7308:
7296:
7295:
7276:
7269:
7265:
7261:
7259:
7257:
7256:
7251:
7248:
7243:
7225:
7223:
7221:
7220:
7215:
7212:
7207:
7189:
7187:
7185:
7184:
7179:
7176:
7171:
7153:
7151:
7149:
7148:
7143:
7140:
7135:
7108:
7106:
7104:
7103:
7098:
7095:
7090:
7072:
7060:
7048:
7041:
7039:
7037:
7036:
7031:
7028:
7023:
7005:
6998:
6991:
6979:
6972:
6968:
6964:
6957:
6953:
6949:
6945:
6941:
6937:
6906:
6890:
6879:
6872:
6863:(2) < … <
6853:
6816:-element subset
6815:
6807:
6782:
6773:
6764:
6755:
6746:
6737:
6733:
6724:
6720:
6712:
6703:
6694:
6685:
6682:such that every
6681:
6677:
6668:
6664:
6656:
6652:
6643:
6639:
6635:
6631:
6613:
6609:
6600:
6596:
6588:
6584:
6573:
6563:
6556:
6545:
6543:
6542:
6537:
6525:
6523:
6522:
6517:
6505:
6503:
6502:
6497:
6485:
6483:
6482:
6477:
6465:
6463:
6462:
6457:
6445:
6443:
6442:
6437:
6425:
6423:
6422:
6417:
6406:(the subsets of
6405:
6403:
6402:
6397:
6395:
6394:
6372:
6370:
6369:
6364:
6340:Ramsey's theorem
6324:
6317:
6313:
6309:
6305:
6276:
6265:
6248:
6241:
6224:
6210:
6206:
6202:
6187:
6175:
6171:
6167:
6163:
6159:
6141:
6130:
6109:
6105:
6101:
6064:
6053:
6049:
6042:
6038:
6034:
6030:
5999:
5960:
5935:
5928:
5917:
5910:
5906:
5902:
5885:
5881:
5877:
5866:
5851:
5829:
5815:
5789:
5781:
5774:
5770:
5756:
5740:
5716:
5702:
5698:
5697:
5696:
5692:
5686:
5682:
5677:
5664:
5643:
5639:
5628:
5624:
5620:
5616:
5612:
5608:
5603:
5579:
5570:vertex colorings
5563:
5559:
5538:
5536:
5535:
5530:
5525:
5524:
5523:
5522:
5521:
5520:
5474:
5473:
5470:
5451:
5447:
5433:
5429:
5425:
5397:
5385:
5381:
5377:
5373:
5369:
5348:
5331:
5327:
5320:such that it is
5319:
5315:
5307:
5303:
5299:
5295:
5251:
5249:
5248:
5243:
5241:
5240:
5216:
5215:
5202:
5186:
5184:
5183:
5178:
5176:
5175:
5151:
5150:
5137:
5121:
5119:
5118:
5113:
5111:
5110:
5086:
5085:
5076:
5075:
5042:
5041:
5017:
5016:
5008:
4998:
4982:
4980:
4979:
4974:
4953:
4951:
4950:
4945:
4940:
4938:
4937:
4936:
4908:
4907:
4892:
4890:
4889:
4856:
4854:
4853:
4852:
4851:
4849:
4835:
4830:
4825:
4814:
4794:
4793:
4788:
4777:
4771:
4765:
4747:
4732:
4698:
4684:
4682:
4681:
4676:
4671:
4667:
4653:
4630:
4622:
4621:
4619:
4618:
4612:
4609:
4599:
4585:
4583:
4582:
4577:
4565:
4563:
4562:
4557:
4555:
4554:
4529:
4527:
4526:
4521:
4519:
4518:
4490:
4488:
4487:
4482:
4477:
4476:
4475:
4466:
4454:
4447:
4446:
4395:
4392:
4390:
4389:
4314:
4312:
4311:
4306:
4301:
4300:
4291:
4290:
4268:
4213:
4212:
4204:
4198:
4193:
4189:
4184:
4181:
4145:
4144:
4143:
4136:
4128:
4118:
4116:
4115:
4110:
4105:
4104:
4100:
4087:
4085:
4081:
4076:
4070:
4010:
4008:
4007:
4002:
3997:
3988:
3987:
3986:
3971:
3914:
3893:
3891:
3890:
3885:
3880:
3879:
3878:
3872:
3861:
3843:
3805:
3389:
3382:
3376:
3360:
3357:
3319:Dynamic Survey 1
3313:
3309:
3295:
3284:
3277:
3262:
3251:
3234:
3227:
3220:
3209:
3191:
3187:
3183:
3173:
3155:
3145:
3138:
3130:
3126:
3122:
3118:
3104:, and therefore
3103:
3081:
3077:
3073:
3059:
3052:
3048:
3034:
3027:
3007:computation time
2993:
2989:
2985:
2970:
2964:
2958:
2949:
2947:
2946:
2943:
2940:
2932:
2921:
2912:
2905:
2877:
2870:
2863:
2851:
2836:
2829:
2814:
2810:
2798:
2783:
2753:
2749:
2745:
2741:
2737:
2733:
2714:
2710:
2706:
2691:
2676:
2653:
2632:
2628:
2610:
2600:
2596:
2579:
2571:
2547:
2518:
2506:
2502:
2492:
2490:
2489:
2484:
2473:
2472:
2460:
2459:
2435:
2434:
2410:
2409:
2387:
2379:
2375:
2368:
2364:
2362:
2361:
2356:
2348:
2347:
2335:
2334:
2310:
2309:
2285:
2284:
2257:
2255:
2254:
2249:
2238:
2237:
2225:
2224:
2200:
2199:
2175:
2174:
2153:
2152:
2134:
2133:
2111:
2093:
2091:
2090:
2085:
2050:
2042:
2030:
2023:
2011:
2004:
2001:Then either the
2000:
1998:
1997:
1992:
1954:
1946:
1934:
1932:
1931:
1926:
1915:
1907:
1895:
1893:
1892:
1887:
1879:
1871:
1859:
1852:
1850:
1849:
1844:
1842:
1834:
1822:
1820:
1819:
1814:
1803:
1795:
1783:
1781:
1780:
1775:
1758:
1750:
1734:
1732:
1731:
1726:
1724:
1723:
1699:
1691:
1679:
1677:
1676:
1671:
1669:
1668:
1656:
1648:
1636:
1632:
1628:
1619:
1612:
1608:
1606:
1605:
1600:
1597:
1596:
1586:
1581:
1554:
1550:
1543:
1539:
1524:
1505:
1480:
1478:
1477:
1472:
1382:
1367:
1349:
1345:
1338:
1336:
1335:
1330:
1306:
1294:
1290:
1283:
1279:
1277:
1276:
1271:
1239:
1231:
1219:
1217:
1216:
1211:
1182:
1174:
1162:
1160:
1159:
1154:
1146:
1138:
1130:
1122:
1056:
1048:
1044:
1040:
1032:
1028:
1024:
1020:
1016:
1009:
1005:
972:
970:
969:
964:
874:
859:
844:
829:
807:
803:
776:
765:, respectively.
764:
755:
739:
717:
708:
698:
691:
687:
683:
679:
675:
671:
664:
660:
656:
652:
648:
644:
637:
630:
617:
608:
594:
585:
560:
550:
543:
536:
527:
515:
506:
498:
494:
490:
486:
482:
474:
466:
462:
458:
443:
437:, and therefore
436:
427:
415:
407:
399:
391:
387:
383:
379:
367:
363:
356:
347:
267:
254:
241:
234:
230:
219:
215:
211:
207:
184:
163:
147:
121:
117:
113:
99:vertices. (Here
98:
94:
90:
71:
52:
48:
29:, in one of its
27:Ramsey's theorem
10845:
10844:
10840:
10839:
10838:
10836:
10835:
10834:
10815:
10814:
10765:
10762:
10648:
10630:
10573:
10544:
10510:
10422:
10326:
10321:
10320:
10285:
10281:
10258:10.2307/2272866
10238:
10234:
10221:
10217:
10200:
10196:
10188:
10186:
10178:
10174:
10137:
10130:
10119:
10115:
10108:
10094:
10090:
10056:10.2307/2371374
10040:
10036:
10028:
10015:
10013:"Ramsey Theory"
10011:Gould, Martin.
10009:
10005:
9998:
9949:
9945:
9897:
9893:
9845:
9841:
9810:
9806:
9799:
9775:
9771:
9719:
9715:
9667:
9660:
9632:
9626:
9622:
9610:
9604:
9600:
9586:
9582:
9571:
9564:
9553:
9546:
9532:
9525:
9516:
9514:
9506:
9505:
9501:
9493:
9491:
9489:
9463:
9459:
9451:Quanta Magazine
9442:
9438:
9405:
9401:
9348:
9344:
9307:
9303:
9294:
9292:
9276:
9273:
9272:
9268:
9220:
9216:
9179:
9175:
9167:Quanta Magazine
9158:
9154:
9137:
9133:
9116:
9112:
9063:
9052:
9035:
9031:
9021:
9019:
9013:
9009:
8981:
8975:
8971:
8902:
8899:
8898:
8895:
8891:
8862:
8859:
8855:
8829:
8819:
8812:
8763:
8759:
8704:
8700:
8693:
8689:
8683:
8659:Joel H. Spencer
8656:
8652:
8645:"Ramsey Graphs"
8643:
8642:
8635:
8626:
8625:
8621:
8605:
8599:
8595:
8563:
8554:
8549:
8548:
8539:
8535:
8521:independent set
8516:
8512:
8498:
8494:
8489:
8451:Ramsey cardinal
8447:
8428:
8412:
8391:
8366:
8361:
8343:
8340:
8339:
8319:
8316:
8315:
8312:Ramsey cardinal
8308:Justin T. Moore
8290:
8285:
8275:
8271:
8259:
8255:
8253:
8250:
8249:
8232:
8225:
8221:
8220:
8210:
8206:
8194:
8190:
8188:
8185:
8184:
8159:
8154:
8144:
8140:
8128:
8124:
8122:
8119:
8118:
8097:
8092:
8082:
8078:
8064:
8060:
8059:
8055:
8053:
8050:
8049:
8032:
8028:
8026:
8023:
8022:
8009:for all finite
7993:
7988:
7978:
7974:
7962:
7958:
7956:
7953:
7952:
7949:
7943:
7930:
7923:
7916:
7909:
7902:
7895:
7888:
7881:
7863:
7855:
7840:
7833:
7826:
7822:
7811:
7796:
7794:Directed graphs
7784:
7777:
7770:
7763:
7756:
7746:
7742:
7738:
7735:
7731:
7727:
7723:
7719:
7715:
7706:
7700:
7690:
7686:
7681:
7675:
7665:
7661:
7655:
7649:
7645:
7641:
7637:
7633:
7626:
7610:
7587:
7582:
7581:
7579:
7576:
7575:
7572:
7563:
7560:
7556:
7553:
7544:
7541:
7537:
7511:
7506:
7493:
7488:
7475:
7471:
7462:
7458:
7456:
7453:
7452:
7406:
7398:
7396:
7392:
7384:
7378:
7374:
7369:
7367:
7364:
7363:
7357:
7353:
7327:
7322:
7309:
7304:
7291:
7287:
7285:
7282:
7281:
7274:
7267:
7263:
7244:
7239:
7233:
7230:
7229:
7227:
7208:
7203:
7197:
7194:
7193:
7191:
7172:
7167:
7161:
7158:
7157:
7155:
7136:
7125:
7119:
7116:
7115:
7113:
7091:
7086:
7080:
7077:
7076:
7074:
7071:
7062:
7059:
7050:
7047:
7043:
7024:
7019:
7013:
7010:
7009:
7007:
7004:
7000:
6993:
6990:
6981:
6977:
6970:
6966:
6963:
6959:
6955:
6951:
6947:
6943:
6939:
6935:
6928:
6905:
6892:
6881:
6874:
6855:
6851:
6837:
6827:
6817:
6809:
6805:
6798:
6791:
6784:
6781:
6775:
6772:
6766:
6763:
6757:
6754:
6748:
6745:
6739:
6735:
6732:
6726:
6722:
6714:
6711:
6705:
6702:
6696:
6693:
6687:
6683:
6679:
6676:
6670:
6666:
6658:
6654:
6651:
6645:
6641:
6637:
6633:
6629:
6615:
6611:
6608:
6602:
6598:
6590:
6586:
6575:
6565:
6558:
6554:
6531:
6528:
6527:
6511:
6508:
6507:
6491:
6488:
6487:
6471:
6468:
6467:
6451:
6448:
6447:
6431:
6428:
6427:
6411:
6408:
6407:
6390:
6386:
6378:
6375:
6374:
6358:
6355:
6354:
6336:
6334:Infinite graphs
6331:
6319:
6315:
6311:
6307:
6298:
6284:
6278:
6267:
6259:
6250:
6243:
6232:
6226:
6217:
6212:
6208:
6204:
6200:
6185:
6182:
6173:
6169:
6165:
6161:
6149:
6143:
6136:
6120:
6114:
6107:
6103:
6087:
6072:
6066:
6055:
6051:
6048:
6044:
6040:
6036:
6032:
6027:
6021:
6014:
6007:
6001:
5997:
5991:
5984:
5978:
5975:
5967:
5965:Generalizations
5951:
5945:
5930:
5923:
5912:
5908:
5904:
5893:
5887:
5883:
5879:
5875:
5857:
5841:
5835:
5823:
5817:
5797:
5791:
5787:
5779:
5772:
5764:
5758:
5754:
5738:
5731:
5710:
5704:
5700:
5694:
5690:
5689:
5688:
5684:
5680:
5667:
5658:
5652:
5644:-vertex graph.
5641:
5634:
5626:
5622:
5618:
5614:
5610:
5606:
5593:
5577:
5561:
5553:
5547:
5501:
5497:
5496:
5492:
5491:
5487:
5469:
5465:
5463:
5460:
5459:
5449:
5441:
5435:
5431:
5427:
5423:
5395:
5392:
5383:
5379:
5375:
5371:
5359:
5353:
5342:
5336:
5329:
5325:
5317:
5313:
5305:
5301:
5297:
5293:
5290:
5266:complete graphs
5258:
5233:
5229:
5211:
5207:
5198:
5192:
5189:
5188:
5168:
5164:
5146:
5142:
5133:
5127:
5124:
5123:
5103:
5099:
5081:
5077:
5071:
5067:
5034:
5030:
5007:
5003:
4994:
4988:
4985:
4984:
4962:
4959:
4958:
4926:
4922:
4909:
4897:
4893:
4891:
4885:
4881:
4839:
4834:
4815:
4813:
4812:
4808:
4795:
4778:
4776:
4772:
4770:
4761:
4755:
4752:
4751:
4734:
4728:
4689:
4652:
4648:
4643:
4640:
4639:
4628:
4613:
4610:
4605:
4604:
4602:
4601:
4590:
4571:
4568:
4567:
4547:
4543:
4535:
4532:
4531:
4511:
4507:
4499:
4496:
4495:
4471:
4456:
4450:
4449:
4448:
4421:
4417:
4393: and
4391:
4385:
4381:
4346:
4343:
4342:
4296:
4292:
4264:
4242:
4238:
4203:
4199:
4183:
4182:
4180:
4154:
4151:
4150:
4141:
4139:
4138:
4130:
4123:
4096:
4092:
4088:
4075:
4074:
4069:
4022:
4019:
4018:
3976:
3972:
3970:
3923:
3920:
3919:
3906:
3874:
3862:
3845:
3839:
3838:
3837:
3814:
3811:
3810:
3768:
3767:The inequality
3765:
3401:
3396:
3378:
3363:
3332:
3311:
3297:
3286:
3279:
3264:
3253:
3238:
3229:
3222:
3211:
3196:
3189:
3185:
3178:
3168:
3150:
3140:
3136:
3128:
3124:
3120:
3105:
3090:
3079:
3075:
3061:
3054:
3050:
3036:
3029:
3022:
3019:
2991:
2987:
2976:
2966:
2960:
2944:
2941:
2938:
2937:
2935:
2934:
2931:
2927:
2922:
2917:
2907:
2900:
2893:
2876:
2872:
2869:
2865:
2862:
2853:
2838:
2834:
2816:
2812:
2808:
2785:
2758:
2751:
2747:
2743:
2739:
2735:
2716:
2712:
2708:
2693:
2678:
2663:
2660:
2650:
2644:
2634:
2630:
2615:
2609:
2608:
2602:
2598:
2595:
2594:
2581:
2573:
2568:
2562:
2549:
2546:
2543:
2537:
2520:
2508:
2504:
2501:
2500:
2494:
2468:
2464:
2449:
2445:
2424:
2420:
2405:
2401:
2393:
2390:
2389:
2381:
2377:
2370:
2366:
2343:
2339:
2324:
2320:
2299:
2295:
2280:
2276:
2268:
2265:
2264:
2233:
2229:
2214:
2210:
2189:
2185:
2170:
2166:
2148:
2144:
2129:
2125:
2117:
2114:
2113:
2106:
2100:
2046:
2038:
2036:
2033:
2032:
2029:
2025:
2022:
2013:
2010:
2006:
2002:
1950:
1942:
1940:
1937:
1936:
1911:
1903:
1901:
1898:
1897:
1875:
1867:
1865:
1862:
1861:
1854:
1838:
1830:
1828:
1825:
1824:
1799:
1791:
1789:
1786:
1785:
1754:
1746:
1744:
1741:
1740:
1719:
1715:
1695:
1687:
1685:
1682:
1681:
1664:
1660:
1652:
1644:
1642:
1639:
1638:
1634:
1630:
1627:
1621:
1618:
1614:
1610:
1592:
1588:
1582:
1571:
1564:
1561:
1560:
1552:
1549:
1545:
1541:
1526:
1507:
1488:
1391:
1388:
1387:
1369:
1354:
1347:
1344:
1340:
1312:
1309:
1308:
1305:
1296:
1292:
1289:
1285:
1281:
1235:
1227:
1225:
1222:
1221:
1178:
1170:
1168:
1165:
1164:
1142:
1134:
1126:
1118:
1062:
1059:
1058:
1050:
1046:
1042:
1034:
1030:
1026:
1022:
1018:
1014:
1007:
980:
886:
883:
882:
861:
846:
831:
809:
805:
795:
788:
783:
775:
769:
763:
757:
754:
748:
738:
732:
716:
710:
703:
693:
689:
685:
681:
677:
673:
666:
662:
658:
654:
650:
646:
642:
632:
625:
622:
621:
620:
619:
616:
610:
607:
601:
597:
596:
595:
587:
586:
575:
555:
545:
538:
537:, showing that
535:
529:
526:
520:
514:
508:
500:
496:
492:
488:
484:
476:
468:
464:
460:
456:
453:double counting
438:
435:
429:
426:
420:
409:
401:
393:
389:
385:
381:
380:, to vertices,
377:
368:and so (by the
365:
361:
355:
349:
346:
340:
331:of that colour.
294:
293:
282:
274:
262:
256:
249:
243:
236:
232:
229:
225:
217:
213:
209:
205:
196:
186:
181:
175:
165:
161:
155:
149:
145:
119:
115:
100:
96:
92:
77:
58:
50:
46:
31:graph-theoretic
17:
12:
11:
5:
10843:
10833:
10832:
10827:
10813:
10812:
10806:
10800:
10791:
10781:
10761:
10760:External links
10758:
10757:
10756:
10712:(13): 130505,
10701:
10675:
10651:
10646:
10633:
10628:
10615:
10606:Spencer, J. H.
10598:
10577:
10571:
10530:
10500:
10488:(4): 292–294,
10474:
10445:math/0607788v1
10438:(2): 941–960,
10425:
10420:
10407:
10372:(2): 291–336,
10361:
10349:(3): 354–360,
10325:
10322:
10319:
10318:
10279:
10252:(3): 387–390.
10232:
10215:
10194:
10172:
10128:
10113:
10106:
10088:
10050:(3): 600–610.
10034:
10031:on 2022-01-30.
10003:
9996:
9943:
9920:(2): 153–196.
9904:Sudakov, Benny
9891:
9839:
9826:(2): 324–333.
9804:
9797:
9769:
9746:(5): 513–535.
9730:Sudakov, Benny
9713:
9674:Sudakov, Benny
9658:
9645:(3): 373–404.
9620:
9598:
9580:
9562:
9544:
9523:
9499:
9487:
9457:
9436:
9399:
9362:(2): 291–336.
9342:
9321:(2): 221–293.
9301:
9266:
9253:10.1.1.46.5058
9246:(3): 173–207,
9214:
9193:(3): 354–360.
9173:
9152:
9131:
9110:
9050:
9029:
9007:
8994:(2): 193–209.
8969:
8927:
8924:
8921:
8918:
8915:
8912:
8909:
8906:
8889:
8853:
8842:(3): 309–322.
8810:
8757:
8698:
8687:
8681:
8650:
8633:
8619:
8593:
8551:
8550:
8547:
8546:
8533:
8515:-clique or an
8491:
8490:
8488:
8485:
8484:
8483:
8478:
8473:
8468:
8463:
8458:
8453:
8446:
8443:
8426:
8423:David Seetapun
8410:
8390:
8387:
8372:
8369:
8364:
8360:
8356:
8353:
8350:
8347:
8336:large cardinal
8323:
8293:
8288:
8284:
8278:
8274:
8270:
8267:
8262:
8258:
8235:
8228:
8224:
8219:
8213:
8209:
8205:
8202:
8197:
8193:
8162:
8157:
8153:
8147:
8143:
8139:
8136:
8131:
8127:
8100:
8095:
8091:
8085:
8081:
8077:
8074:
8067:
8063:
8058:
8035:
8031:
7996:
7991:
7987:
7981:
7977:
7973:
7970:
7965:
7961:
7945:Main article:
7942:
7939:
7795:
7792:
7788:(5, 5; 3) ≥ 88
7781:(4, 6; 3) ≥ 63
7774:(4, 5; 3) ≥ 35
7767:(4, 4; 3) = 13
7733:
7722:between 1 and
7704:
7698:
7679:
7673:
7659:
7653:
7625:
7622:
7596:
7593:
7590:
7585:
7567:
7558:
7548:
7539:
7534:
7533:
7522:
7519:
7514:
7509:
7505:
7501:
7496:
7491:
7487:
7483:
7478:
7474:
7470:
7465:
7461:
7446:
7445:
7430:
7427:
7424:
7421:
7418:
7415:
7412:
7409:
7404:
7401:
7395:
7391:
7387:
7381:
7377:
7372:
7355:
7350:
7349:
7338:
7335:
7330:
7325:
7321:
7317:
7312:
7307:
7303:
7299:
7294:
7290:
7247:
7242:
7238:
7211:
7206:
7202:
7175:
7170:
7166:
7139:
7134:
7131:
7128:
7124:
7094:
7089:
7085:
7066:
7054:
7045:
7027:
7022:
7018:
7002:
6985:
6961:
6927:
6924:
6896:
6842:
6832:
6822:
6803:
6796:
6789:
6779:
6770:
6761:
6752:
6743:
6730:
6725:consisting of
6709:
6700:
6691:
6674:
6649:
6627:
6606:
6548:
6547:
6535:
6515:
6495:
6475:
6455:
6435:
6415:
6393:
6389:
6385:
6382:
6362:
6335:
6332:
6330:
6327:
6296:
6282:
6254:
6230:
6215:
6181:
6178:
6147:
6118:
6085:
6070:
6046:
6025:
6019:
6012:
6005:
5995:
5989:
5982:
5974:
5971:
5966:
5963:
5949:
5938:counting lemma
5891:
5839:
5821:
5795:
5762:
5730:
5727:
5708:
5656:
5551:
5540:
5539:
5528:
5519:
5516:
5513:
5510:
5507:
5504:
5500:
5495:
5490:
5486:
5483:
5480:
5477:
5468:
5439:
5416:towers of twos
5391:
5388:
5357:
5340:
5296:be a graph on
5289:
5286:
5257:
5256:Induced Ramsey
5254:
5239:
5236:
5232:
5228:
5225:
5222:
5219:
5214:
5210:
5205:
5201:
5197:
5174:
5171:
5167:
5163:
5160:
5157:
5154:
5149:
5145:
5140:
5136:
5132:
5109:
5106:
5102:
5098:
5095:
5092:
5089:
5084:
5080:
5074:
5070:
5066:
5063:
5060:
5057:
5054:
5051:
5048:
5045:
5040:
5037:
5033:
5029:
5026:
5023:
5020:
5014:
5011:
5006:
5001:
4997:
4993:
4972:
4969:
4966:
4955:
4954:
4943:
4935:
4932:
4929:
4925:
4921:
4918:
4915:
4912:
4906:
4903:
4900:
4896:
4888:
4884:
4880:
4877:
4874:
4871:
4868:
4865:
4862:
4859:
4848:
4845:
4842:
4838:
4833:
4828:
4824:
4821:
4818:
4811:
4807:
4804:
4801:
4798:
4791:
4787:
4784:
4781:
4775:
4768:
4764:
4760:
4686:
4685:
4674:
4670:
4665:
4662:
4659:
4656:
4651:
4647:
4575:
4553:
4550:
4546:
4542:
4539:
4517:
4514:
4510:
4506:
4503:
4492:
4491:
4480:
4474:
4469:
4465:
4462:
4459:
4453:
4445:
4442:
4439:
4436:
4433:
4430:
4427:
4424:
4420:
4416:
4413:
4410:
4407:
4404:
4401:
4398:
4388:
4384:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4359:
4356:
4353:
4350:
4316:
4315:
4304:
4299:
4295:
4289:
4286:
4283:
4280:
4277:
4274:
4271:
4267:
4263:
4260:
4257:
4254:
4251:
4248:
4245:
4241:
4237:
4234:
4231:
4228:
4225:
4222:
4219:
4216:
4210:
4207:
4202:
4196:
4192:
4187:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4158:
4120:
4119:
4108:
4103:
4099:
4095:
4091:
4084:
4079:
4073:
4068:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
4029:
4026:
4012:
4011:
4000:
3994:
3991:
3985:
3982:
3979:
3975:
3969:
3966:
3963:
3960:
3957:
3954:
3951:
3948:
3945:
3942:
3939:
3936:
3933:
3930:
3927:
3895:
3894:
3883:
3877:
3871:
3868:
3865:
3860:
3857:
3854:
3851:
3848:
3842:
3836:
3833:
3830:
3827:
3824:
3821:
3818:
3764:
3761:
3758:
3757:
3752:
3750:
3748:
3746:
3744:
3742:
3740:
3738:
3736:
3734:
3730:
3729:
3726:
3721:
3719:
3717:
3715:
3713:
3711:
3709:
3707:
3705:
3701:
3700:
3697:
3694:
3689:
3687:
3685:
3683:
3681:
3679:
3677:
3675:
3671:
3670:
3667:
3664:
3661:
3656:
3654:
3652:
3650:
3648:
3646:
3644:
3640:
3639:
3636:
3633:
3630:
3627:
3622:
3620:
3618:
3616:
3614:
3612:
3608:
3607:
3604:
3601:
3598:
3595:
3592:
3587:
3585:
3583:
3581:
3579:
3575:
3574:
3571:
3568:
3565:
3562:
3559:
3556:
3551:
3549:
3547:
3545:
3541:
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3514:
3512:
3510:
3506:
3505:
3502:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3476:
3474:
3470:
3469:
3466:
3463:
3460:
3457:
3454:
3451:
3448:
3445:
3442:
3437:
3433:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3397:
3392:
3149:The fact that
3102:(3, 3) − 1 = 9
3018:
3015:
2929:
2915:
2892:
2889:
2874:
2867:
2857:
2659:
2658:Ramsey numbers
2656:
2648:
2642:
2606:
2604:
2589:
2585:
2566:
2557:
2541:
2532:
2524:
2498:
2496:
2482:
2479:
2476:
2471:
2467:
2463:
2458:
2455:
2452:
2448:
2444:
2441:
2438:
2433:
2430:
2427:
2423:
2419:
2416:
2413:
2408:
2404:
2400:
2397:
2354:
2351:
2346:
2342:
2338:
2333:
2330:
2327:
2323:
2319:
2316:
2313:
2308:
2305:
2302:
2298:
2294:
2291:
2288:
2283:
2279:
2275:
2272:
2247:
2244:
2241:
2236:
2232:
2228:
2223:
2220:
2217:
2213:
2209:
2206:
2203:
2198:
2195:
2192:
2188:
2184:
2181:
2178:
2173:
2169:
2165:
2162:
2159:
2156:
2151:
2147:
2143:
2140:
2137:
2132:
2128:
2124:
2121:
2099:
2096:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2049:
2045:
2041:
2027:
2017:
2008:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1953:
1949:
1945:
1924:
1921:
1918:
1914:
1910:
1906:
1885:
1882:
1878:
1874:
1870:
1841:
1837:
1833:
1812:
1809:
1806:
1802:
1798:
1794:
1773:
1770:
1767:
1764:
1761:
1757:
1753:
1749:
1722:
1718:
1714:
1711:
1708:
1705:
1702:
1698:
1694:
1690:
1667:
1663:
1659:
1655:
1651:
1647:
1625:
1616:
1595:
1591:
1585:
1580:
1577:
1574:
1570:
1547:
1482:
1481:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1342:
1328:
1325:
1322:
1319:
1316:
1300:
1287:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1238:
1234:
1230:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1181:
1177:
1173:
1152:
1149:
1145:
1141:
1137:
1133:
1129:
1125:
1121:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
974:
973:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
929:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
893:
890:
787:
784:
782:
779:
773:
761:
752:
736:
714:
707:(3, 3, 3) = 17
697:(3, 3, 3) ≤ 17
636:(3, 3, 4) = 30
629:(3, 3, 3) = 17
614:
605:
599:
598:
589:
588:
580:
579:
578:
577:
576:
574:
573:(3, 3, 3) = 17
567:
533:
524:
512:
433:
424:
353:
344:
311:, if the edge
288:2-colour case
281:
275:
273:
270:
260:
247:
227:
216:between 1 and
201:
194:
179:
173:
159:
153:
74:complete graph
43:complete graph
39:edge labelling
15:
9:
6:
4:
3:
2:
10842:
10831:
10828:
10826:
10825:Ramsey theory
10823:
10822:
10820:
10810:
10807:
10804:
10801:
10799:
10797:
10792:
10789:
10785:
10782:
10778:
10774:
10773:
10768:
10764:
10763:
10753:
10749:
10745:
10741:
10737:
10733:
10729:
10725:
10720:
10715:
10711:
10707:
10702:
10697:
10692:
10688:
10684:
10680:
10676:
10672:
10668:
10664:
10660:
10656:
10655:Ramsey, F. P.
10652:
10649:
10647:0-201-02787-9
10643:
10639:
10634:
10631:
10625:
10621:
10616:
10611:
10610:Ramsey Theory
10607:
10603:
10599:
10595:
10591:
10587:
10583:
10578:
10574:
10568:
10564:
10560:
10556:
10552:
10551:
10543:
10539:
10535:
10531:
10528:
10524:
10520:
10516:
10509:
10505:
10501:
10496:
10491:
10487:
10483:
10479:
10475:
10471:
10467:
10463:
10459:
10455:
10451:
10446:
10441:
10437:
10433:
10432:
10426:
10423:
10417:
10413:
10408:
10405:
10401:
10397:
10393:
10389:
10385:
10380:
10375:
10371:
10367:
10366:Invent. Math.
10362:
10357:
10352:
10348:
10344:
10340:
10336:
10335:Komlós, János
10332:
10331:Ajtai, Miklós
10328:
10327:
10314:
10310:
10306:
10302:
10298:
10294:
10290:
10283:
10275:
10271:
10267:
10263:
10259:
10255:
10251:
10247:
10243:
10236:
10228:
10227:
10219:
10210:
10205:
10198:
10185:
10184:
10176:
10168:
10164:
10159:
10154:
10150:
10146:
10142:
10135:
10133:
10124:
10117:
10109:
10103:
10099:
10092:
10083:
10079:
10075:
10071:
10066:
10061:
10057:
10053:
10049:
10045:
10038:
10027:
10023:
10022:
10014:
10007:
9999:
9993:
9988:
9983:
9978:
9973:
9969:
9965:
9964:Thomas, Robin
9961:
9957:
9956:Rödl, Vojtěch
9953:
9952:Conlon, David
9947:
9938:
9933:
9928:
9923:
9919:
9915:
9914:
9913:Combinatorica
9909:
9905:
9901:
9895:
9886:
9881:
9876:
9871:
9867:
9863:
9862:
9857:
9853:
9849:
9848:Conlon, David
9843:
9834:
9829:
9825:
9821:
9820:
9815:
9808:
9800:
9794:
9789:
9784:
9780:
9773:
9764:
9759:
9754:
9749:
9745:
9741:
9740:
9739:Combinatorica
9735:
9731:
9727:
9723:
9722:Conlon, David
9717:
9708:
9703:
9698:
9693:
9689:
9685:
9684:
9679:
9675:
9671:
9665:
9663:
9653:
9648:
9644:
9640:
9639:
9638:Combinatorica
9631:
9624:
9616:
9609:
9602:
9594:
9590:
9584:
9576:
9569:
9567:
9558:
9551:
9549:
9540:
9536:
9530:
9528:
9513:
9509:
9503:
9490:
9484:
9480:
9476:
9472:
9468:
9461:
9453:
9452:
9447:
9440:
9432:
9428:
9423:
9418:
9414:
9410:
9403:
9395:
9391:
9387:
9383:
9379:
9375:
9370:
9365:
9361:
9357:
9353:
9346:
9338:
9334:
9329:
9324:
9320:
9316:
9312:
9305:
9291:
9287:
9283:
9279:
9270:
9263:
9259:
9254:
9249:
9245:
9241:
9237:
9233:
9229:
9225:
9218:
9210:
9206:
9201:
9196:
9192:
9188:
9184:
9177:
9169:
9168:
9163:
9156:
9147:
9142:
9135:
9126:
9121:
9114:
9105:
9104:10.37236/5254
9100:
9096:
9092:
9087:
9082:
9078:
9074:
9073:
9068:
9061:
9059:
9057:
9055:
9045:
9040:
9033:
9018:
9011:
9002:
8997:
8993:
8989:
8988:
8980:
8973:
8965:
8961:
8956:
8951:
8947:
8943:
8942:
8925:
8922:
8916:
8913:
8910:
8904:
8893:
8885:
8881:
8877:
8873:
8872:
8865:
8857:
8849:
8845:
8841:
8837:
8836:
8828:
8826:
8817:
8815:
8806:
8802:
8798:
8794:
8790:
8786:
8781:
8776:
8773:(3): 032301.
8772:
8768:
8761:
8753:
8749:
8745:
8741:
8737:
8733:
8728:
8723:
8719:
8715:
8714:
8709:
8702:
8696:
8691:
8684:
8678:
8674:
8670:
8666:
8665:
8660:
8654:
8646:
8640:
8638:
8629:
8623:
8616:(5): 306–312.
8615:
8611:
8604:
8597:
8588:
8583:
8579:
8575:
8571:
8567:
8561:
8559:
8557:
8552:
8544:of the graph.
8543:
8542:automorphisms
8537:
8530:
8526:
8522:
8510:
8506:
8502:
8496:
8492:
8482:
8479:
8477:
8474:
8472:
8469:
8467:
8464:
8462:
8459:
8457:
8454:
8452:
8449:
8448:
8442:
8440:
8439:Kőnig's lemma
8436:
8435:Kőnig's lemma
8432:
8424:
8420:
8417:, one of the
8416:
8408:
8404:
8400:
8396:
8386:
8370:
8367:
8362:
8354:
8345:
8337:
8321:
8313:
8309:
8291:
8286:
8276:
8265:
8260:
8233:
8226:
8211:
8200:
8195:
8182:
8178:
8160:
8155:
8145:
8134:
8129:
8117:implies that
8116:
8098:
8093:
8083:
8072:
8065:
8056:
8033:
8020:
8016:
8012:
7994:
7989:
7979:
7963:
7948:
7938:
7934:
7926:
7919:
7912:
7905:
7898:
7891:
7884:
7878:
7876:
7872:
7867:
7861:
7851:
7847:
7843:
7837:
7830:
7818:
7814:
7809:
7805:
7801:
7791:
7787:
7780:
7773:
7766:
7759:
7753:
7749:
7711:
7707:
7697:
7693:
7682:
7672:
7668:
7662:
7652:
7631:
7621:
7619:
7614:
7591:
7570:
7566:
7551:
7547:
7520:
7517:
7512:
7507:
7503:
7499:
7494:
7489:
7485:
7481:
7476:
7472:
7468:
7463:
7459:
7451:
7450:
7449:
7428:
7422:
7419:
7416:
7410:
7407:
7402:
7399:
7393:
7389:
7379:
7375:
7362:
7361:
7360:
7359:is finite as
7336:
7333:
7328:
7323:
7319:
7315:
7310:
7305:
7301:
7297:
7292:
7288:
7280:
7279:
7278:
7271:
7245:
7240:
7236:
7209:
7204:
7200:
7173:
7168:
7164:
7137:
7132:
7129:
7126:
7122:
7110:
7092:
7087:
7083:
7069:
7065:
7057:
7053:
7025:
7020:
7016:
6996:
6988:
6984:
6974:
6933:
6923:
6921:
6917:
6913:
6908:
6903:
6899:
6895:
6888:
6884:
6877:
6870:
6866:
6862:
6858:
6849:
6845:
6841:
6835:
6831:
6825:
6821:
6813:
6802:
6795:
6788:
6778:
6769:
6760:
6751:
6742:
6729:
6718:
6708:
6699:
6690:
6673:
6662:
6648:
6626:
6622:
6618:
6605:
6594:
6582:
6578:
6572:
6568:
6561:
6552:
6533:
6513:
6493:
6473:
6453:
6433:
6413:
6391:
6383:
6360:
6352:
6349:
6348:
6347:
6346:terminology.
6345:
6344:set-theoretic
6341:
6326:
6322:
6303:
6299:
6292:
6288:
6281:
6274:
6270:
6263:
6257:
6253:
6246:
6240:
6236:
6229:
6222:
6218:
6197:
6195:
6191:
6177:
6157:
6153:
6146:
6140:
6134:
6128:
6124:
6117:
6111:
6099:
6095:
6091:
6084:
6080:
6076:
6069:
6063:
6059:
6028:
6018:
6011:
6004:
5998:
5988:
5981:
5970:
5962:
5959:
5955:
5948:
5943:
5939:
5933:
5926:
5921:
5920:tower of twos
5918:growing as a
5915:
5901:
5897:
5890:
5872:
5870:
5864:
5860:
5855:
5849:
5845:
5838:
5833:
5827:
5820:
5813:
5809:
5805:
5801:
5794:
5785:
5776:
5771:is linear in
5768:
5761:
5752:
5748:
5744:
5735:
5729:Special cases
5726:
5722:
5720:
5714:
5707:
5679:
5675:
5671:
5662:
5655:
5650:
5645:
5637:
5632:
5613:and suitable
5605:
5601:
5597:
5590:
5586:
5581:
5575:
5571:
5567:
5557:
5550:
5545:
5526:
5514:
5508:
5505:
5502:
5498:
5493:
5488:
5484:
5478:
5466:
5458:
5457:
5456:
5453:
5445:
5438:
5421:
5417:
5413:
5410:, Deuber and
5409:
5405:
5401:
5387:
5367:
5363:
5356:
5350:
5346:
5339:
5335:
5323:
5311:
5285:
5283:
5280:, Deuber and
5279:
5275:
5271:
5267:
5263:
5253:
5237:
5234:
5226:
5223:
5220:
5212:
5208:
5203:
5199:
5195:
5172:
5169:
5161:
5158:
5155:
5147:
5143:
5138:
5134:
5130:
5107:
5104:
5096:
5093:
5090:
5082:
5078:
5072:
5068:
5064:
5058:
5055:
5052:
5046:
5043:
5038:
5035:
5027:
5024:
5021:
5012:
5009:
5004:
4999:
4995:
4991:
4970:
4967:
4964:
4941:
4933:
4930:
4927:
4919:
4916:
4913:
4904:
4901:
4898:
4894:
4886:
4882:
4878:
4872:
4869:
4866:
4860:
4857:
4846:
4843:
4840:
4836:
4831:
4826:
4822:
4819:
4816:
4805:
4802:
4799:
4789:
4785:
4782:
4779:
4773:
4766:
4762:
4758:
4750:
4749:
4748:
4745:
4741:
4737:
4731:
4726:
4722:
4718:
4714:
4710:
4706:
4702:
4696:
4692:
4672:
4668:
4663:
4660:
4657:
4654:
4649:
4638:
4637:
4636:
4634:
4626:
4617:
4608:
4597:
4593:
4587:
4573:
4551:
4548:
4544:
4540:
4537:
4515:
4512:
4508:
4504:
4501:
4478:
4467:
4463:
4460:
4457:
4440:
4434:
4431:
4428:
4425:
4422:
4418:
4414:
4408:
4405:
4402:
4396:
4386:
4378:
4375:
4372:
4366:
4360:
4357:
4354:
4348:
4341:
4340:
4339:
4337:
4333:
4329:
4325:
4321:
4302:
4297:
4293:
4284:
4281:
4278:
4275:
4272:
4265:
4258:
4255:
4252:
4249:
4243:
4239:
4235:
4229:
4226:
4223:
4217:
4214:
4208:
4205:
4200:
4194:
4190:
4185:
4171:
4165:
4162:
4159:
4149:
4148:
4147:
4134:
4129:, this gives
4126:
4106:
4101:
4097:
4093:
4089:
4082:
4077:
4071:
4060:
4054:
4051:
4048:
4042:
4036:
4033:
4030:
4024:
4017:
4016:
4015:
3998:
3992:
3989:
3983:
3980:
3977:
3973:
3961:
3955:
3952:
3949:
3943:
3937:
3934:
3931:
3925:
3918:
3917:
3916:
3913:
3909:
3904:
3900:
3881:
3869:
3866:
3863:
3858:
3855:
3852:
3849:
3846:
3834:
3828:
3825:
3822:
3816:
3809:
3808:
3807:
3803:
3799:
3795:
3791:
3787:
3783:
3779:
3775:
3771:
3756:
3753:
3751:
3749:
3747:
3745:
3743:
3741:
3739:
3737:
3735:
3732:
3731:
3727:
3725:
3722:
3720:
3718:
3716:
3714:
3712:
3710:
3708:
3706:
3703:
3702:
3698:
3695:
3693:
3690:
3688:
3686:
3684:
3682:
3680:
3678:
3676:
3673:
3672:
3668:
3665:
3662:
3660:
3657:
3655:
3653:
3651:
3649:
3647:
3645:
3642:
3641:
3637:
3634:
3631:
3628:
3626:
3623:
3621:
3619:
3617:
3615:
3613:
3610:
3609:
3605:
3602:
3599:
3596:
3593:
3591:
3588:
3586:
3584:
3582:
3580:
3577:
3576:
3572:
3569:
3566:
3563:
3560:
3557:
3555:
3552:
3550:
3548:
3546:
3543:
3542:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3518:
3515:
3513:
3511:
3508:
3507:
3503:
3500:
3497:
3494:
3491:
3488:
3485:
3482:
3480:
3477:
3475:
3472:
3471:
3467:
3464:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3441:
3438:
3435:
3434:
3400:
3395:
3391:
3390:
3386:
3381:
3374:
3370:
3366:
3359:
3355:
3351:
3347:
3343:
3339:
3335:
3330:
3326:
3325:
3320:
3315:
3308:
3304:
3300:
3293:
3289:
3285:are given by
3282:
3275:
3271:
3267:
3260:
3256:
3249:
3245:
3241:
3236:
3232:
3225:
3218:
3214:
3207:
3203:
3199:
3193:
3181:
3175:
3171:
3165:
3163:
3159:
3158:Brendan McKay
3153:
3147:
3143:
3134:
3116:
3112:
3108:
3101:
3097:
3093:
3088:
3083:
3072:
3068:
3064:
3057:
3053:: a graph on
3047:
3043:
3039:
3032:
3025:
3014:
3012:
3008:
3004:
3000:
2995:
2983:
2979:
2974:
2969:
2963:
2956:
2952:
2920:
2914:
2910:
2903:
2898:
2888:
2885:
2884:Brendan McKay
2881:
2864:with no blue
2860:
2856:
2849:
2845:
2841:
2831:
2827:
2823:
2819:
2806:
2802:
2796:
2792:
2788:
2781:
2777:
2773:
2769:
2765:
2761:
2755:
2731:
2727:
2723:
2719:
2704:
2700:
2696:
2689:
2685:
2681:
2674:
2670:
2666:
2655:
2651:
2641:
2637:
2626:
2622:
2618:
2612:
2592:
2588:
2584:
2577:
2569:
2560:
2556:
2552:
2544:
2535:
2531:
2527:
2523:
2516:
2512:
2480:
2469:
2465:
2461:
2456:
2453:
2450:
2446:
2439:
2436:
2431:
2428:
2425:
2421:
2417:
2414:
2411:
2406:
2402:
2395:
2385:
2373:
2344:
2340:
2336:
2331:
2328:
2325:
2321:
2314:
2311:
2306:
2303:
2300:
2296:
2292:
2289:
2286:
2281:
2277:
2270:
2262:
2258:
2245:
2234:
2230:
2226:
2221:
2218:
2215:
2211:
2204:
2201:
2196:
2193:
2190:
2186:
2182:
2179:
2176:
2171:
2167:
2160:
2157:
2149:
2145:
2141:
2138:
2135:
2130:
2126:
2119:
2109:
2104:
2095:
2078:
2075:
2072:
2069:
2066:
2060:
2057:
2054:
2051:
2043:
2020:
2016:
1988:
1982:
1979:
1976:
1973:
1970:
1964:
1961:
1958:
1955:
1947:
1922:
1919:
1916:
1908:
1883:
1880:
1872:
1857:
1835:
1810:
1807:
1804:
1796:
1771:
1768:
1765:
1762:
1759:
1751:
1738:
1720:
1716:
1712:
1709:
1706:
1703:
1700:
1692:
1665:
1661:
1657:
1649:
1624:
1593:
1589:
1583:
1578:
1575:
1572:
1568:
1558:
1544:vertices. If
1537:
1533:
1529:
1522:
1518:
1514:
1510:
1503:
1499:
1495:
1491:
1486:
1468:
1465:
1459:
1456:
1453:
1450:
1447:
1441:
1438:
1432:
1429:
1426:
1423:
1420:
1414:
1411:
1405:
1402:
1399:
1393:
1386:
1385:
1384:
1380:
1376:
1372:
1365:
1361:
1357:
1351:
1323:
1317:
1314:
1303:
1299:
1267:
1261:
1258:
1255:
1252:
1249:
1243:
1240:
1232:
1204:
1201:
1198:
1195:
1192:
1186:
1183:
1175:
1150:
1147:
1139:
1131:
1123:
1115:
1109:
1106:
1103:
1100:
1097:
1091:
1088:
1082:
1079:
1076:
1073:
1070:
1064:
1054:
1041:is blue, and
1038:
1013:
1003:
999:
995:
991:
987:
983:
978:
960:
954:
951:
948:
945:
942:
936:
933:
927:
924:
921:
918:
915:
909:
906:
900:
897:
894:
888:
881:
878:
877:
876:
872:
868:
864:
857:
853:
849:
842:
838:
834:
828:
824:
820:
816:
812:
802:
798:
793:
786:2-colour case
778:
772:
766:
760:
751:
745:
743:
742:Clebsch graph
735:
727:
726:Clebsch graph
723:
719:
713:
706:
700:
696:
669:
639:
635:
628:
613:
604:
593:
584:
572:
566:
564:
558:
552:
548:
542:(3, 3) > 5
541:
532:
523:
517:
511:
504:
480:
472:
454:
449:
447:
441:
432:
423:
419:
413:
405:
397:
375:
371:
352:
343:
337:
330:
326:
322:
318:
314:
310:
306:
302:
298:
291:
286:
279:
269:
266:
259:
253:
246:
239:
223:
204:
200:
193:
189:
182:
172:
168:
162:
152:
142:
140:
136:
132:
131:Ramsey theory
128:
123:
111:
107:
103:
88:
84:
80:
75:
69:
65:
61:
56:
44:
40:
36:
32:
28:
24:
23:combinatorics
19:
10795:
10770:
10709:
10705:
10686:
10682:
10662:
10658:
10638:Graph Theory
10637:
10619:
10609:
10585:
10581:
10554:
10548:
10518:
10514:
10485:
10481:
10435:
10429:
10411:
10369:
10365:
10346:
10342:
10299:(1): 37–42.
10296:
10292:
10282:
10249:
10245:
10235:
10225:
10218:
10197:
10187:, retrieved
10182:
10175:
10148:
10144:
10122:
10116:
10098:Graph Theory
10097:
10091:
10047:
10043:
10037:
10026:the original
10019:
10006:
9967:
9946:
9917:
9911:
9894:
9865:
9859:
9842:
9823:
9822:. Series B.
9817:
9807:
9778:
9772:
9743:
9737:
9716:
9687:
9681:
9642:
9636:
9623:
9614:
9601:
9592:
9583:
9574:
9556:
9538:
9515:. Retrieved
9511:
9502:
9492:, retrieved
9470:
9460:
9449:
9439:
9412:
9408:
9402:
9359:
9355:
9345:
9318:
9314:
9304:
9293:. Retrieved
9289:
9281:
9277:
9269:
9243:
9239:
9235:
9231:
9227:
9223:
9217:
9190:
9186:
9176:
9165:
9155:
9134:
9113:
9076:
9070:
9032:
9020:. Retrieved
9010:
8991:
8990:. Series B.
8985:
8972:
8955:1703.08768v2
8945:
8939:
8892:
8878:(1): 97–98.
8875:
8869:
8863:
8856:
8839:
8833:
8824:
8770:
8766:
8760:
8717:
8711:
8701:
8690:
8663:
8653:
8622:
8613:
8609:
8596:
8577:
8573:
8536:
8511:, either an
8509:simple graph
8501:Brualdi 2010
8495:
8402:
8398:
8392:
8014:
8010:
7950:
7932:
7924:
7917:
7910:
7903:
7896:
7889:
7882:
7879:
7874:
7870:
7865:
7859:
7849:
7845:
7841:
7838:
7828:
7816:
7812:
7804:P. Erdős
7799:
7797:
7785:
7778:
7771:
7764:
7757:
7754:
7747:
7709:
7702:
7695:
7691:
7677:
7670:
7666:
7657:
7650:
7627:
7615:
7568:
7564:
7549:
7545:
7535:
7447:
7351:
7272:
7111:
7067:
7063:
7055:
7051:
6994:
6986:
6982:
6975:
6929:
6909:
6901:
6897:
6893:
6886:
6882:
6875:
6868:
6864:
6860:
6856:
6847:
6843:
6839:
6833:
6829:
6823:
6819:
6811:
6800:
6793:
6786:
6776:
6767:
6758:
6749:
6740:
6738:elements of
6727:
6716:
6706:
6697:
6688:
6671:
6660:
6646:
6624:
6620:
6616:
6603:
6592:
6580:
6576:
6570:
6566:
6559:
6550:
6549:
6350:
6339:
6337:
6320:
6301:
6294:
6290:
6286:
6279:
6272:
6268:
6261:
6255:
6251:
6244:
6238:
6234:
6227:
6220:
6213:
6198:
6193:
6189:
6183:
6155:
6151:
6144:
6138:
6133:tower of two
6126:
6122:
6115:
6112:
6097:
6093:
6089:
6082:
6078:
6074:
6067:
6061:
6057:
6023:
6016:
6009:
6002:
5993:
5986:
5979:
5976:
5968:
5957:
5953:
5946:
5931:
5924:
5922:with height
5913:
5899:
5895:
5888:
5873:
5862:
5858:
5847:
5843:
5836:
5825:
5818:
5811:
5807:
5803:
5799:
5792:
5777:
5766:
5759:
5736:
5732:
5723:
5712:
5705:
5673:
5669:
5660:
5653:
5646:
5635:
5599:
5595:
5582:
5555:
5548:
5541:
5454:
5443:
5436:
5393:
5365:
5361:
5354:
5351:
5344:
5337:
5333:
5310:induced copy
5291:
5259:
4956:
4743:
4739:
4735:
4729:
4699:is given by
4694:
4690:
4687:
4615:
4606:
4595:
4591:
4588:
4493:
4332:Sahasrabudhe
4317:
4132:
4124:
4121:
4013:
3911:
3907:
3896:
3801:
3797:
3793:
3789:
3785:
3781:
3777:
3773:
3769:
3766:
3754:
3723:
3691:
3658:
3624:
3589:
3553:
3516:
3478:
3439:
3398:
3393:
3372:
3368:
3364:
3353:
3349:
3345:
3341:
3337:
3333:
3322:
3318:
3316:
3306:
3302:
3298:
3291:
3287:
3280:
3273:
3269:
3265:
3258:
3254:
3247:
3243:
3239:
3237:
3230:
3223:
3216:
3212:
3205:
3201:
3197:
3194:
3179:
3176:
3169:
3166:
3151:
3148:
3141:
3114:
3110:
3106:
3099:
3095:
3091:
3084:
3070:
3066:
3062:
3055:
3045:
3041:
3037:
3030:
3023:
3020:
3017:Known values
2996:
2981:
2977:
2967:
2961:
2954:
2950:
2924:
2919:Joel Spencer
2908:
2901:
2895:
2880:Ramsey graph
2879:
2858:
2854:
2847:
2843:
2839:
2832:
2825:
2821:
2817:
2794:
2790:
2786:
2779:
2775:
2771:
2767:
2763:
2759:
2756:
2729:
2725:
2721:
2717:
2702:
2698:
2694:
2687:
2683:
2679:
2672:
2668:
2664:
2662:The numbers
2661:
2646:
2639:
2635:
2624:
2620:
2616:
2613:
2601:-monochrome
2590:
2586:
2582:
2580:-monochrome
2575:
2564:
2558:
2554:
2550:
2539:
2533:
2529:
2525:
2521:
2514:
2510:
2383:
2371:
2260:
2259:
2107:
2102:
2101:
2018:
2014:
1855:
1853:is even and
1622:
1535:
1531:
1527:
1520:
1516:
1512:
1508:
1501:
1497:
1493:
1489:
1484:
1483:
1378:
1374:
1370:
1363:
1359:
1355:
1352:
1301:
1297:
1052:
1036:
1001:
997:
993:
989:
985:
981:
976:
975:
879:
870:
866:
862:
855:
851:
847:
840:
836:
832:
826:
822:
818:
814:
810:
800:
796:
789:
770:
767:
758:
749:
746:
733:
730:
711:
704:
702:To see that
701:
694:
667:
640:
633:
626:
623:
611:
602:
570:
556:
553:
546:
539:
530:
521:
518:
509:
502:
478:
470:
450:
439:
430:
421:
417:
411:
403:
395:
359:
350:
341:
328:
324:
320:
316:
312:
308:
304:
300:
296:
277:
264:
257:
251:
244:
237:
202:
198:
191:
187:
177:
170:
166:
157:
150:
143:
134:
127:Frank Ramsey
124:
109:
105:
101:
86:
82:
78:
67:
63:
59:
26:
20:
18:
10784:Ramsey@Home
10689:: 108–115,
10679:Spencer, J.
10665:: 264–286,
10557:: 463–470,
10534:Erdős, Paul
10521:: 125–132,
10478:Erdős, Paul
9927:0707.4159v2
8948:(1): 5–13.
8827:(4,5) = 25"
8587:10.37236/21
8505:Harary 1972
8476:Ramsey game
7873:is the two
7630:hypergraphs
7624:Hypergraphs
6965:denote the
6920:cardinality
6526:subsets of
6225:by letting
6180:Hypergraphs
5973:More colors
5940:for sparse
5874:For graphs
5832:superlinear
5683:with small
5631:Paley graph
5609:with small
3763:Asymptotics
3182:(5, 5) = 43
3154:(4, 5) = 25
3144:(4, 4) = 18
3133:Paley graph
3131:graph (the
3117:(3, 4) ≤ 18
3011:exponential
2973:brute force
2031:. The case
1339:has a blue
1295:has a blue
137:, that is,
53:be any two
10819:Categories
10602:Graham, R.
10324:References
10209:2011.00683
10189:2020-06-02
10125:: 304–308.
9977:1601.01493
9900:Fox, Jacob
9868:: 206–29.
9852:Fox, Jacob
9726:Fox, Jacob
9670:Fox, Jacob
9517:2023-07-12
9494:2023-06-27
9422:2306.04007
9295:2023-06-27
9146:2303.09521
9125:2310.17099
9086:1504.02403
9044:2401.00392
8780:1510.01884
8727:1511.04206
8671:, p.
8525:Gross 2008
7875:directions
6585:. Given a
6194:hypergraph
6135:of height
6106:copies of
6081:) :=
5869:degenerate
5544:Kohayakawa
5420:Paul Erdős
5322:isomorphic
3377:(sequence
3190:(5, 5, 43)
3186:(5, 5, 42)
3129:(4, 4, 17)
3121:(4, 4, 16)
3033:(4, 2) = 4
3026:(3, 3) = 6
2803:using the
2801:Paul Erdős
1487:. Suppose
1284:has a red
670:(3, 3) = 6
559:(3, 3) ≤ 6
549:(3, 3) = 6
497:6 × 6 = 36
442:(3, 3) ≤ 6
280:(3, 3) = 6
10777:EMS Press
10719:1201.1842
10588:: 97–98,
10504:Erdős, P.
10379:0908.0429
10313:1432-0665
10266:1943-5886
10167:1715-0868
9875:1204.6645
9753:1002.0045
9697:0706.4112
9589:Erdős, P.
9535:Erdős, P.
9394:1432-1297
9369:0908.0429
9328:1302.5963
9248:CiteSeerX
9209:0097-3165
9022:17 August
8923:≤
8805:118724989
8720:: 15023.
8371:ω
8355:κ
8349:→
8346:κ
8322:κ
8273:ℵ
8266:↛
8257:ℵ
8223:ℵ
8208:ℵ
8201:↛
8192:ℵ
8142:ℵ
8135:↛
8126:ℵ
8080:ℵ
8073:↛
8062:ℵ
8030:ℵ
7976:ℵ
7969:→
7960:ℵ
7521:⋯
7518:∩
7500:∩
7482:∩
7420:−
7390:≤
7337:⋯
7334:⊇
7316:⊇
7298:⊇
7006:. Define
6859:(1) <
6000:, define
5647:In 2010,
5485:≤
5288:Statement
5235:−
5224:
5170:−
5159:
5105:−
5094:
5065:≤
5044:≤
5036:−
5025:
4931:−
4917:
4902:−
4879:≤
4858:≤
4844:−
4832:−
4803:
4709:Szemerédi
4661:
4646:Θ
4574:ε
4549:−
4538:δ
4513:−
4502:ε
4426:δ
4423:−
4415:≤
4379:ε
4376:−
4367:≤
4282:
4276:
4256:
4244:−
4236:≤
4215:≤
4043:≥
3990:π
3981:−
3944:≤
3867:−
3856:−
3835:≤
3755:798–17730
3728:581–9797
3699:343–4432
3696:329–2683
3669:292–2134
3666:252–1379
3164:in 1995.
3137:2.46 × 10
3113:(4, 3) +
3109:(4, 4) ≤
3098:(4, 2) +
3094:(4, 3) ≤
3009:is still
2507:for some
2454:−
2429:−
2415:…
2329:−
2304:−
2290:…
2219:−
2194:−
2180:…
2158:≤
2139:…
2076:−
2052:≥
1974:−
1956:≥
1917:≥
1881:≥
1805:≥
1766:−
1760:≥
1739:, either
1713:−
1707:−
1569:∑
1466:−
1457:−
1424:−
1412:≤
1318:∪
1259:−
1241:≥
1196:−
1184:≥
1107:−
1074:−
952:−
919:−
907:≤
792:induction
493:2 × 3 = 6
489:1 × 4 = 4
485:0 × 5 = 0
224:of order
10744:24116761
10608:(1990),
10540:(1935),
9966:(eds.).
9906:(2009).
9732:(2012).
9676:(2008).
9079:(3): 3.
8661:(1994),
8568:(2011).
8445:See also
7935:(7) ≤ 47
7927:(6) = 28
7920:(5) = 14
7880:We have
7800:directed
7061:. Since
6976:For any
6653:to each
6614:and let
6426:of size
6351:Theorem.
6242:and for
6160:, where
5204:′
5139:′
5000:′
4767:′
4631:-vertex
3903:Szekeres
3724:565–5366
3692:282–1532
3663:219–840
3638:204–949
3635:183–656
3632:134–427
3629:115–273
3606:149–381
3603:133–282
3600:101–194
3125:6.4 × 10
3049:for all
2916:—
2103:Lemma 2.
1935:Suppose
1033:if edge
880:Lemma 1.
272:Examples
222:subgraph
10779:, 2001
10752:1303361
10724:Bibcode
10527:0168494
10470:9238219
10462:2552114
10404:2429894
10384:Bibcode
10274:2272866
10082:0004862
10074:2371374
9617:: 1–17.
9374:Bibcode
9091:Bibcode
8785:Bibcode
8752:2992738
8732:Bibcode
8523:, see (
8503:) and (
8334:, is a
7913:(4) = 8
7906:(3) = 4
7899:(2) = 2
7892:(1) = 1
7885:(0) = 0
7871:colours
7810:). Let
7260:
7228:
7224:
7192:
7188:
7156:
7152:
7114:
7107:
7075:
7040:
7008:
5834:(i.e.
5693:⁄
5589:Sudakov
5576:graphs
5332:is the
4725:Keevash
4620:
4603:
4320:Spencer
4318:due to
4140:√
3659:205–497
3625:102–161
3597:80–133
3573:92–136
3570:73–106
3383:in the
3380:A212954
3321:of the
3192:graph.
3069:, 2) ≥
3044:, 2) =
2994:nodes.
2948:
2936:
2112:, then
1307:and so
875:exist.
817:, 2) =
139:subsets
37:in any
35:cliques
10750:
10742:
10644:
10626:
10569:
10525:
10468:
10460:
10418:
10402:
10311:
10272:
10264:
10165:
10104:
10080:
10072:
9994:
9795:
9485:
9392:
9250:
9207:
8866:(5, 5)
8803:
8750:
8679:
8540:Up to
8527:) or (
7929:, and
7154:is in
6958:. Let
6601:, let
6137:~ log
6065:. Let
5846:) = ω(
5678:-graph
5649:Conlon
5629:, the
5604:-graph
5574:degree
5404:Hajnal
5274:Hajnal
4721:Bohman
4719:, and
4717:Morris
4707:, and
4705:Komlós
4627:in an
4328:Morris
4324:Conlon
4131:101 ≤
3594:59–85
3567:59–79
3564:49–58
3561:36–40
3539:40–41
3283:< 3
3233:(8, 8)
3226:(6, 6)
3219:> 5
3172:(5, 5)
2911:(6, 6)
2904:(5, 5)
2261:Proof.
2110:> 2
1823:Since
1045:is in
1029:is in
977:Proof.
10786:is a
10748:S2CID
10714:arXiv
10545:(PDF)
10511:(PDF)
10466:S2CID
10440:arXiv
10400:S2CID
10374:arXiv
10270:JSTOR
10204:arXiv
10151:(2).
10070:JSTOR
10029:(PDF)
10016:(PDF)
9972:arXiv
9922:arXiv
9870:arXiv
9748:arXiv
9692:arXiv
9633:(PDF)
9611:(PDF)
9417:arXiv
9415:(2).
9364:arXiv
9323:arXiv
9234:/log
9141:arXiv
9120:arXiv
9081:arXiv
9039:arXiv
9017:"DS1"
8982:(PDF)
8950:arXiv
8830:(PDF)
8801:S2CID
8775:arXiv
8748:S2CID
8722:arXiv
8606:(PDF)
8487:Notes
7931:34 ≤
7701:, …,
7676:, …,
7656:, …,
7632:. An
6854:with
6838:, …,
6806:, …}
6551:Proof
6446:) in
6271:≥ 3,
6264:) = 2
6203:be a
6190:graph
6158:) ≤ 2
6096:, …,
6039:into
6022:, …,
5992:, …,
5944:that
5782:is a
5743:cycle
5741:is a
5715:) ≤ 2
5663:) ≤ 2
5558:) ≤ 2
5446:) ≤ 2
5400:Erdős
5270:Erdős
4701:Ajtai
4494:with
3899:Erdős
3788:− 1,
3590:43–48
3327:, by
3294:) = 1
3278:with
3252:with
3210:with
2975:) is
2897:Erdős
2645:, …,
2597:or a
2519:or a
1500:− 1,
1485:Proof
1362:− 1,
988:− 1,
854:− 1,
781:Proof
242:(and
197:, …,
176:, …,
156:, …,
10794:The
10740:PMID
10642:ISBN
10624:ISBN
10567:ISBN
10416:ISBN
10309:ISSN
10262:ISSN
10163:ISSN
10102:ISBN
9992:ISBN
9793:ISBN
9483:ISBN
9390:ISSN
9205:ISSN
9024:2023
8677:ISBN
8669:SIAM
8578:1000
8368:<
8013:and
7852:; 2)
7808:1964
7783:and
7755:For
7683:; m)
7644:and
6871:+ 1)
6850:+ 1)
6814:+ 1)
6734:and
6719:+ 1)
6663:+ 1)
6595:+ 1)
6353:Let
6314:and
6293:) ≤
6237:) =
6199:Let
6168:and
6056:1 ≤
5956:) ≤
5898:) ≤
5802:) =
5784:tree
5751:star
5747:path
5587:and
5412:Rödl
5408:Pósa
5406:and
5292:Let
5282:Rödl
5278:Pósa
5276:and
4957:For
4723:and
4693:(3,
4614:log
4594:(3,
4530:and
4336:book
4330:and
4322:and
4127:= 10
3901:and
3804:− 1)
3792:) +
3780:) ≤
3385:OEIS
3344:) =
3305:) =
3301:(2,
3296:and
3290:(1,
3261:≤ 10
3228:and
3195:For
3160:and
2986:for
2957:− 1)
2933:has
2837:for
2811:and
2770:) =
2750:and
2578:− 1)
2509:1 ≤
2386:− 1)
2376:and
1680:and
1633:and
1523:− 1)
1506:and
1381:− 1)
1368:and
1017:and
1012:sets
1004:− 1)
992:) +
873:− 1)
860:and
825:) =
821:(2,
631:and
388:and
307:and
255:and
118:and
49:and
10732:doi
10710:111
10691:doi
10667:doi
10590:doi
10559:doi
10490:doi
10450:doi
10436:170
10392:doi
10370:181
10351:doi
10301:doi
10254:doi
10153:doi
10060:hdl
10052:doi
9982:doi
9932:doi
9880:doi
9866:256
9828:doi
9783:doi
9758:doi
9702:doi
9688:219
9647:doi
9475:doi
9427:doi
9413:199
9382:doi
9360:181
9333:doi
9280:(3,
9258:doi
9238:",
9226:(3,
9195:doi
9099:doi
8996:doi
8960:doi
8938:".
8880:doi
8868:".
8844:doi
8793:doi
8740:doi
8582:doi
8413:of
8393:In
8181:ZFC
7760:= 3
7750:= 2
6997:+ 1
6878:(1)
6836:(2)
6826:(1)
6774:of
6756:in
6678:of
6623:\ {
6583:+ 1
6562:= 1
6486:of
6323:= 3
6283:ind
6275:≥ 2
6247:≥ 1
6148:ind
6119:ind
6110:).
6086:ind
6071:ind
6006:ind
5950:ind
5934:(Δ)
5927:(Δ)
5916:(Δ)
5892:ind
5871:).
5840:ind
5830:is
5822:ind
5810:log
5796:ind
5786:on
5778:If
5763:ind
5753:on
5749:or
5737:If
5709:ind
5676:,λ)
5657:ind
5638:≥ 2
5633:on
5602:,λ)
5585:Fox
5552:ind
5471:ind
5440:ind
5430:on
5358:ind
5341:ind
5324:to
5312:of
5221:log
5156:log
5091:log
5022:log
4914:log
4800:log
4713:Kim
4658:log
4635:is
4279:log
4273:log
4253:log
3733:10
3558:25
3536:36
3533:28
3530:23
3527:18
3524:14
3504:10
3431:10
3358:.)
3058:− 1
2593:− 1
2561:− 1
2536:− 1
2517:− 2
2374:− 1
2105:If
2021:– 1
1896:or
1858:– 1
1784:or
1538:− 1
1304:− 1
1220:or
1049:if
794:on
503:xyz
418:any
329:rst
319:or
268:).
240:= 2
122:.)
76:on
21:In
10821::
10775:,
10769:,
10746:,
10738:,
10730:,
10722:,
10708:,
10687:18
10685:,
10663:30
10661:,
10586:13
10584:,
10565:,
10553:,
10547:,
10536:;
10523:MR
10517:,
10513:,
10486:53
10484:,
10464:,
10458:MR
10456:,
10448:,
10434:,
10398:,
10390:,
10382:,
10368:,
10347:29
10345:,
10337:;
10333:;
10307:.
10297:46
10295:.
10291:.
10268:.
10260:.
10250:42
10248:.
10244:.
10161:.
10149:13
10147:.
10143:.
10131:^
10078:MR
10076:.
10068:.
10058:.
10048:63
10046:.
10018:.
9990:.
9980:.
9962:;
9930:.
9918:29
9916:.
9910:.
9902:;
9878:.
9864:.
9858:.
9850:;
9824:66
9816:.
9791:.
9756:.
9744:32
9742:.
9736:.
9728:;
9724:;
9700:.
9686:.
9680:.
9672:;
9661:^
9643:18
9641:.
9635:.
9613:.
9565:^
9547:^
9526:^
9510:.
9481:,
9469:,
9448:.
9425:.
9411:.
9388:.
9380:.
9372:.
9358:.
9354:.
9331:.
9319:58
9317:.
9313:.
9288:.
9256:,
9242:,
9203:.
9191:29
9189:.
9185:.
9164:.
9097:.
9089:.
9077:22
9075:.
9069:.
9053:^
8992:69
8984:.
8958:.
8946:89
8944:.
8926:48
8876:13
8874:.
8840:19
8838:.
8832:.
8813:^
8799:.
8791:.
8783:.
8771:93
8769:.
8746:.
8738:.
8730:.
8716:.
8710:.
8675:,
8667:,
8636:^
8614:33
8612:.
8608:.
8580:.
8576:.
8555:^
8431:ZF
8314:,
8306:.
8183:,
8175:.
8017:.
7937:.
7922:,
7915:,
7908:,
7901:,
7894:,
7887:,
7864:≥
7860:un
7848:,
7827:≥
7790:.
7776:,
7708:;
7571:+1
7552:+1
7277:,
7270:.
7266:,
7109:.
7070:+1
7058:+1
6989:+1
6973:.
6942:,
6938:,
6828:,
6799:,
6792:,
6630:}.
6621:X
6619:=
6579:=
6569:≤
6325:.
6302:ck
6277:,
6258:+1
6249:,
6176:.
6092:,
6060:≤
6015:,
5985:,
5958:cn
5900:cn
5775:.
5745:,
5721:.
5580:.
5402:,
5386:.
5349:.
5272:,
5252:.
4742:,
4703:,
4586:.
4545:50
3915:,
3910:=
3800:,
3776:,
3704:9
3674:8
3643:7
3611:6
3578:5
3554:18
3544:4
3521:9
3509:3
3501:9
3498:8
3495:7
3492:6
3489:5
3486:4
3483:3
3473:2
3468:1
3465:1
3462:1
3459:1
3456:1
3453:1
3450:1
3447:1
3444:1
3436:1
3428:9
3425:8
3422:7
3419:6
3416:5
3413:4
3410:3
3407:2
3404:1
3387:)
3371:,
3352:,
3340:,
3314:.
3272:,
3257:,
3246:,
3215:,
3204:,
3146:.
2882:.
2861:−1
2846:,
2830:.
2824:,
2793:,
2778:,
2766:,
2754:.
2728:,
2720:=
2701:,
2686:,
2671:,
2563:,
2538:,
2513:≤
1559:,
1534:+
1530:=
1519:,
1511:=
1492:=
1469:1.
1377:,
1053:vw
1037:vw
1025:,
1000:,
869:,
839:,
808:,
799:+
774:14
762:16
753:15
744:.
737:16
715:16
699:.
638:.
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