The finite version of Ramsey's theorem says that for positive integers r, k, a_1,... ,a_r, there exists a least number n=R(a_1, \ldots, a_r; k) so that if X is an n-element set and all k-subsets of X are r-coloured, then there exists an i and an a_i-set A so that all k-subsets of A are coloured with the ith colour.In this paper, the bound R(4, 5; 3) &gt;= 35 is shown by using a SAT solver to construct a red--blue colouring of the triples chosen from a 34-element set

Dybizbański, Janusz

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Contributions to Discrete Mathematics (E-Journal, University of Calgary)

Volume 13, Number 2, Pages 112–115ISSN 1715-0868A LOWER BOUND ON THE HYPERGRAPH RAMSEYNUMBER R(4, 5; 3)JANUSZ DYBIZBAN´SKIAbstract. The finite version of Ramsey’s theorem says that for posi-tive integers r, k, a1, . . . , ar, there exists a least number n = R(a1, . . . ,ar; k) so that if X is an n-element set and all k-subsets of X are r-coloured, then there exists an i and an ai-set A so that all k-subsets ofA are coloured with the ith colour.In this paper, the bound R(4, 5; 3) ≥ 35 is shown by using a SATsolver to construct a red–blue colouring of the triples chosen from a34-element set.1. IntroductionIn 1930, Ramsey proved the following theorem:Theorem 1.1 ([11]). Let r, k, a1, . . . , ar be given positive integers. Thenthere is an integer n with the following property. If all k-subsets of an n-setare coloured with r colours, then for some i, 1 ≤ i ≤ r, there exists an ai-setentirely coloured in colour i (all of its k-subsets have colour i).The smallest n for which Ramsey’s theorem holds, we call a Ramseynumber and is denoted by R(a1, . . . , ar; k). This notation is used by thesurvey by Radziszowski [10]. Note that there are at least two other notationsfor these numbers in the literature, namely: Rk(a1, . . . , ar), used for examplein [5], or R(k)(a1, . . . , ar), used in [2]. Since colouring of all k-subsets canbe viewed as colouring of the edges of complete k-uniform hypergraphs,numbers R(a1, . . . , ar; k), for k ≥ 3, are also called hypergraph Ramseynumbers.For k = 2, only ten exact values for nontrivial Ramsey numbers are known(see [10] for details). For k = 3, only one exact nontrivial value is known,namely R(4, 4; 3) = 13, where R(4, 4; 3) ≥ 13 was proved in 1969 by Isbell [7]and equality was shown by McKay and Radziszowski [9] in 1991.In this paper we deal with the number R(4, 5; 3). In 1983 Isbell [8]proved that R(4, 5; 3) ≥ 24, and in 1998 Exoo [4] presented a colouringwhich gives the bound R(4, 5; 3) ≥ 33. Up to the author’s knowledgeReceived by the editors July 27, 2016, and in revised form November 8, 2017.2010 Mathematics Subject Classification. 05C15.Key words and phrases. Ramsey numbers, Hypergraphs.c©2018 University of Calgary112A LOWER BOUND ON THE HYPERGRAPH RAMSEY NUMBER R(4, 5; 3) 113the best upper bound for R(4, 5; 3) can be obtained by using one step ofthe Erdo˝s–Szekeres recursion [3] and known bounds for Ramsey numberswith smaller parameters, see [10]. These give the estimation R(4, 5; 3) ≤R(R(3, 5; 3), R(4, 4; 3); 2) + 1 = R(5, 13; 2) + 1 ≤ 1139. In this paper weshow that R(4, 5; 3) ≥ 35. This bound is shown by producing a 2-colouringof the triples of set with 34-elements. The colouring is formed by dividing alltriples into classes and then by looking through 2-colouring of the classes.Similar approaches were used to establish lower bound for graph Ramseynumbers (k = 2). For example, Harborth and Krause [6] searched throughcolourings such that the colouring matrix is partitioned into cyclic orbits.2. ColouringTheorem 2.1. Let V = {0, . . . , 33}. There exists a colouring c : (V3) →{red, blue}, such that no 4-subset of V is entirely coloured in red, and no5-subset of V is entirely coloured in blue.Proof. This colouring is achieved by first dividing all such triples into 176classes and then by giving a 2-colouring of the classes. For each two integersa, b satisfying 1 ≤ a ≤ 11 and a + 1 ≤ b ≤ 22, define the class:Cab ={{0 + d, a + d, b + d} : 0 ≤ d ≤ 33},where addition is modulo 34. It is easy to see that there are 176 classes,each contains 34 3-sets and the classes are pairwise disjoint. Hence, we havea partition of(V3)into 176 disjoint classes. To find a proper colouring of theclasses we use a SAT solver. We construct a formula with 176 variables—onefor every class Cab. For every E ∈(V3), let f(E) be the variable for the classthat contains E. We assume that E is blue if the variable f(E) is true, andE is red if f(E) is false. We use the following formula:[ ∧S∈(V4)∨E∈(S3)f(E)]∧[ ∧S∈(V5)∨E∈(S3)¬f(E)].The first part of the formula says that in every 4-set at least one of 3-subsetsis coloured in blue (variable is true). Similarly, the second part says that inevery 5-sets at least one of 3-subsets is coloured in red. It may happen thatfor some S ∈ (V4) we have two triples E and E′ such that f(E) = f(E′)and we have two repeated literals in the clause∨E∈(S3)f(E). We simplifythe clauses by removing such repetitions. Similarly it may happen thatfor two sets S and S′ ∈ (V4) the clauses ∨E∈(S3) f(E) and ∨E∈(S′3 ) f(E) areequivalent. We simplify the formula by removing such repetitions. Similarlywe simplify the second part of the formula.Finally, a formula with 176 variables and 9552 clauses is found. We usethe SparrowToRiss [1] SAT solver and find out that the formula is satisfied.The SAT solver finishes in less than five minutes on a personal computer11Computer with processor IntelR© CoreTM i7-4790, 3.60GHz.114 JANUSZ DYBIZBAN´SKIa\b 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 221 0 0 1 0 1 0 1 1 0 1 1 1 1 0 0 0 0 0 1 0 02 0 0 0 0 0 1 1 0 1 0 1 1 1 1 0 1 1 0 0 13 1 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 1 1 14 1 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 15 1 1 0 1 1 1 0 0 0 1 0 0 0 1 1 1 06 0 0 0 0 0 0 1 0 1 0 1 1 0 0 1 07 1 1 0 0 0 0 1 0 0 0 0 0 0 0 18 0 0 1 0 0 1 0 1 1 0 0 1 0 09 0 0 0 0 1 0 0 0 1 0 0 0 010 0 1 0 1 0 1 1 1 0 0 1 111 0 0 0 1 0 0 1 1 0 1 0Figure 1. The colouring of the classes Cab without red 4-subset and blue 5-subset.and returns the assignment that gives the proper colouring of the classespresented in Figure 1.One can easily find, using a computer, that the colouring is proper, i.e.each clique K4 contains at least one 3-set coloured in blue and each cliqueK5 contains at least one 3-set coloured in red. On the website, https://www.inf.ug.edu.pl/ramsey, we posted a simplified formula, satisfyingassignment, the corresponding colouring of all triples, and a C++ programthat verifies that the colouring is proper. AcknowledgementThe author would like to thank the anonymous reviewer for helpful com-ments that contributed to improving the final version of the paper.References1. A. Balint and N. Manthey, Sparrow + CP3 and SparrowToRiss, Proc. of SAT Com-petition 2013: Solver and Benchmark Descriptions (2013), 87–88.2. B. Bollobas, Modern graph theory, Springer, New York, 1998.3. P. Erdo˝s and G. Szekeres, A combinatorial problem in geometry, Compos. Math. 2(1935), 463–470.4. G. Exoo, Ramsey constructions, http://ginger.indstate.edu/ge/RAMSEY/, IndianaState University.5. R. L. Graham, B. L. Rothschild, and J. H. Spencer, Ramsey theory, John Wiley &Sons, 1990.6. H. Harborth and S. Krause, Ramsey numbers for circulant colorings, Congr. Numer.161 (2003), 139–150.7. J. R. Isbell, N(4, 4; 3) ≥ 13, J. Combin. Theory 6 (1969), 210.8. , N(5, 4; 3) ≥ 24, J. Combin. Theory Ser. A 34 (1983), 379–380.9. B. D. McKay and S. P. Radziszowski, The first classical ramsey number for hyper-graphs is computed, Proc. of the Second Annual ACM-SIAM Symposium on DiscreteAlgorithms (San Francisco), SODA 91, 1991, pp. 304–308.10. S. P. Radziszowski, Small Ramsey numbers, Electron. J. Combin. (2014), DynamicSurvey DS1, revision #14, http://www.combinatorics.org.A LOWER BOUND ON THE HYPERGRAPH RAMSEY NUMBER R(4, 5; 3) 11511. F. P. Ramsey, On a problem of formal logic, Proc. Lond. Math. Soc. 30 (1930), 264–286.Institute of Informatics, Faculty of Mathematics, Physics, and Informatics,University of Gdan´sk, 80-308 Gdan´sk, PolandE-mail address: jdybiz@inf.ug.edu.pl

A lower bound on the hypergraph Ramsey number R(4,5;3)

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A lower bound on the hypergraph Ramsey number R(4,5;3)

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