Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-06-23T08:32:06.467Z Has data issue: false hasContentIssue false
  • English
  • Français

Ramsey-type numbers involving graphs and hypergraphs with large girth

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  12 April 2021

Hiêp Hàn ,
Troy Retter ,
Vojtêch Rödl  and
Mathias Schacht
Hiêp Hàn
Affiliation:
Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Chile
Troy Retter
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, USA
Vojtêch Rödl
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, USA
Mathias Schacht*
Affiliation:
Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany
*
*Corresponding author. Email: schacht@math.uni-hamburg.de
Get access Rights & Permissions [Opens in a new window]

Abstract

Erdős asked if, for every pair of positive integers g and k, there exists a graph H having girth (H) = k and the property that every r-colouring of the edges of H yields a monochromatic cycle Ck. The existence of such graphs H was confirmed by the third author and Ruciński.

We consider the related numerical problem of estimating the order of the smallest graph H with this property for given integers r and k. We show that there exists a graph H on R10k2; k15k3 vertices (where R = R(Ck; r) is the r-colour Ramsey number for the cycle Ck) having girth (H) = k and the Ramsey property that every r-colouring of the edges of H yields a monochromatic Ck Two related numerical problems regarding arithmetic progressions in subsets of the integers and cliques in graphs are also considered.

MSC classification

Primary: 05C55: Generalized Ramsey theory
Secondary: 05D10: Ramsey theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 5 , September 2021 , pp. 722 - 740
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

H. Hàn was partly supported by FAPESP (2010/16526-3 and 2013/11353-1).

‡

V. Rödl was supported by NSF grants DMS 1301698 and 1764385.

§

M. Schacht was supported through the Heisenberg-Programme of the DFG.

References

Balogh, J., Morris, R. and Samotij, W. (2015) Independent sets in hypergraphs. J. Amer. Math. Soc. 28 669–709.CrossRefGoogle Scholar
Benson, C. T. (1966) Minimal regular graphs of girths eight and twelve. Canad. J. Math. 18 1091–1094.CrossRefGoogle Scholar
Bloom, T. F. (2016) A quantitative improvement for Roth’s theorem on arithmetic progressions. J. London Math. Soc. (2) 93 643–663.10.1112/jlms/jdw010CrossRefGoogle Scholar
Bondy, J. A. and Erdős, P. (1973) Ramsey numbers for cycles in graphs. J. Combin. Theory Ser. B 14 46–54.CrossRefGoogle Scholar
Bondy, J. A. and Simonovits, M. (1974) Cycles of even length in graphs. J. Combin. Theory Ser. B 16 97–105.CrossRefGoogle Scholar
Erdős, P. (1938) On sequences of integers no one of which divides the product of two others and on some related problems. Mitt. Forsch.-Inst. Math. und Mech. Univ. Tomsk 2 74–82.Google Scholar
Erdős, P. (1964) Extremal problems in graph theory. In Theory of Graphs and its Applications, pp. 29–36. Czechoslovak Academy of Sciences, Prague.Google Scholar
Erdős, P. (1975) Problems and results in combinatorial number theory. Astérisque 24–25 295–310.Google Scholar
Erdős, P. (1975) Problems and results on finite and infinite graphs. In Recent Advances in Graph Theory, pp. 183–192. Academia.Google Scholar
Erdős, P. and Graham, R. L. (1975) On partition theorems for finite graphs. In Infinite and Finite Sets I, Vol. 10 of Colloquia Mathematica Societatis János Bolyai, pp. 515–527. North-Holland.Google Scholar
Erdős, P. and Hajnal, A. (1966) Research problems. Acta Math. Acad. Sci. Hungar 17 61–99.10.1007/BF02020444CrossRefGoogle Scholar
Erdős, P. and Hajnal, A. (1967) Research problems 2–5. J. Combin. Theory 2 104–105.Google Scholar
Erdős, P. and Simonovits, M. (1982) Compactness results in extremal graph theory. Combinatorica 2 275–288.CrossRefGoogle Scholar
Folkman, J. (1970) Graphs with monochromatic complete subgraphs in every edge coloring. SIAM J. Appl. Math. 18 19–24.CrossRefGoogle Scholar
Graham, R. L. and Nešetřil, J. (1986) Large minimal sets which force long arithmetic progressions. J. Combin. Theory Ser. A 42 270–276.CrossRefGoogle Scholar
Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience.10.1002/9781118032718CrossRefGoogle Scholar
Jenssen, M. and Skokan, J. (2016) Exact Ramsey numbers of odd cycles via nonlinear optimisation. arXiv:1608.05705Google Scholar
Nenadov, R. and Steger, A. (2016) A short proof of the random Ramsey theorem. Combin. Probab. Comput. 25 130–144.10.1017/S0963548314000832CrossRefGoogle Scholar
Nešetřil, J. and Rödl, V. (1976) The Ramsey property for graphs with forbidden complete subgraphs. J. Combin. Theory Ser. B 20 243–249.10.1016/0095-8956(76)90015-0CrossRefGoogle Scholar
Nešetřil, J. and Rödl, V. (1976) Vander Waerden theorem for sequences of integers not containing an arithmetic progression of k terms. Comment. Math. Univ. Carolinae 17 675–681.Google Scholar
Nešetřil, J. and Rödl, V. (1981) Simple proof of the existence of restricted Ramsey graphs by means of a partite construction. Combinatorica 1 199–202.CrossRefGoogle Scholar
Nešetřil, J. and Rödl, V. (1984) Sparse Ramsey graphs. Combinatorica 4 71–78.10.1007/BF02579159CrossRefGoogle Scholar
Rödl, V. (1990) On Ramsey families of sets. Graphs Combin. 6 187–195.CrossRefGoogle Scholar
Rödl, V. and Ruciński, A. (1995) Threshold functions for Ramsey properties. J. Amer. Math. Soc. 8 917–942.CrossRefGoogle Scholar
Rödl, V., Ruciński, A. and Schacht, M. (2017) An exponential-type upper bound for Folkman numbers. Combinatorica 37 767–784.CrossRefGoogle Scholar
Sanders, T. (2011) On Roth’s theorem on progressions. Ann. of Math. (2) 174 619–636.CrossRefGoogle Scholar
Saxton, D. and Thomason, A. (2015) Hypergraph containers. Invent. Math. 201 925–992.CrossRefGoogle Scholar
Singleton, R. (1966) On minimal graphs of maximum even girth. J. Combin. Theory 1 306–332.10.1016/S0021-9800(66)80054-6CrossRefGoogle Scholar
Spencer, J. (1975) Restricted Ramsey configurations. J. Combin. Theory Ser. A 19 278–286.CrossRefGoogle Scholar