Hostname: page-component-76c49bb84f-lvxqv Total loading time: 0 Render date: 2025-07-03T17:30:53.391Z Has data issue: false hasContentIssue false
  • English
  • Français

Common graphs with arbitrary chromatic number

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  03 July 2025

Daniel Král͏̌ ,
Jan Volec  and
Fan Wei Open the ORCID record for Fan Wei [Opens in a new window]
Daniel Král͏̌
Affiliation:
Faculty of Informatics, Masaryk University, Botanická 68A, 602 00 Brno, Czech Republic dkral@fi.muni.cz
Jan Volec
Affiliation:
Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Thákurova 9, 160 00 Prague, Czech Republic jan@ucw.cz
Fan Wei
Affiliation:
Duke Mathematics, Campus Box 90320, Durham, NC 27708, USA fan.wei@duke.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Ramsey’s theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. In 1962, Erdős conjectured that the random 2-edge-coloring minimizes the number of monochromatic copies of $K_k$, and the conjecture was extended by Burr and Rosta to all graphs. In the late 1980s, the conjectures were disproved by Thomason and Sidorenko, respectively. A classification of graphs whose number of monochromatic copies is minimized by the random 2-edge-coloring, which are referred to as common graphs, remains a challenging open problem. If Sidorenko’s conjecture, one of the most significant open problems in extremal graph theory, is true, then every 2-chromatic graph is common and, in fact, no 2-chromatic common graph unsettled for Sidorenko’s conjecture is known. While examples of 3-chromatic common graphs were known for a long time, the existence of a 4-chromatic common graph was open until 2012, and no common graph with a larger chromatic number is known.

We construct connected k-chromatic common graphs for every k. This answers a question posed by Hatami et al. [Non-three-colourable common graphs exist, Combin. Probab. Comput. 21 (2012), 734–742], and a problem listed by Conlon et al. [Recent developments in graph Ramsey theory, in Surveys in combinatorics 2015, London Mathematical Society Lecture Note Series, vol. 424 (Cambridge University Press, Cambridge, 2015), 49–118, Problem 2.28]. This also answers in a stronger form the question raised by Jagger et al. [Multiplicities of subgraphs, Combinatorica 16 (1996), 123–131] whether there exists a common graph with chromatic number at least four.

Keywords

Ramsey theorygraph limitschromatic numberspectral methodprobabilistic method

MSC classification

Primary: 05D10: Ramsey theory 05D40: Probabilistic methods 05C15: Coloring of graphs and hypergraphs 05C35: Extremal problems

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 3 , March 2025 , pp. 594 - 634
Check for updates
Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence.

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.)

Article purchase

Temporarily unavailable

References

Alon, N. and Spencer, J. H., The probabilistic method , Wiley Series in Discrete Mathematics and Optimization, fourth edition (John Wiley & Sons, Hoboken, NJ, 2016).Google Scholar
Blakley, G. R. and Roy, P., A Hölder type inequality for symmetric matrices with nonnegative entries, Proc. Amer. Math. Soc. 16 (1965), 1244–1245.Google Scholar
Bradač, D., Sudakov, B. and Wigderson, Y., Counting subgraphs in locally dense graphs, Preprint (2024), arXiv:2406.12418.Google Scholar
Burr, S. A. and Rosta, V., On the Ramsey multiplicities of graphs—problems and recent results , J. Graph Theory 4 (1980), 347–361.10.1002/jgt.3190040403CrossRefGoogle Scholar
Campos, M., Griffiths, S., Morris, R. and Sahasrabudhe, J., An exponential improvement for diagonal ramsey, Preprint (2023), arXiv:2303.09521.Google Scholar
Clark, L., The minimum number of subgraphs in a graph and its complement , J. Graph Theory 16 (1992), 451–458; MR 1185009.10.1002/jgt.3190160506CrossRefGoogle Scholar
Conlon, D., A new upper bound for diagonal Ramsey numbers, Ann. of Math. (2) 170 (2009), 941–960.10.4007/annals.2009.170.941CrossRefGoogle Scholar
Conlon, D., Fox, J. and Sudakov, B., An approximate version of Sidorenko’s conjecture, Geom. Funct. Anal. 20 (2010), 1354–1366.Google Scholar
Conlon, D., Fox, J. and Sudakov, B., Recent developments in graph Ramsey theory, in Surveys in combinatorics 2015, London Mathematical Society Lecture Note Series, vol. 424 (Cambridge University Press, Cambridge, 2015), 49–118; MR 3497267.Google Scholar
Conlon, D., Kim, J. H., Lee, C. and Lee, J., Some advances on Sidorenko’s conjecture, J. Lond. Math. Soc. (2) 98 (2018), 593–608.10.1112/jlms.12142CrossRefGoogle Scholar
Conlon, D. and Lee, J., Finite reflection groups and graph norms , Adv. Math. 315 (2017), 130–165.10.1016/j.aim.2017.05.009CrossRefGoogle Scholar
Conlon, D. and Lee, J., Sidorenko’s conjecture for blow-ups, Discrete Anal. (2021), 2. Au: Please provide volume number for Ref. [CL21], if available.Google Scholar
Cooper, J. W., Král͏̌, D. and Martins, T. L., Finitely forcible graph limits are universal , Adv. Math. 340 (2018), 819–854.10.1016/j.aim.2018.10.019CrossRefGoogle Scholar
Csóka, E., Hubai, T. and Lovász, L., Locally common graphs , J. Graph Theory 102 (2023), 472–483.10.1002/jgt.22881CrossRefGoogle ScholarPubMed
Erdős, P., Graph theory and probability, Canad. J. Math. 11 (1959), 34–38; MR 102081.Google Scholar
Erdős, P., On the number of complete subgraphs contained in certain graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 7 (1962), 459–464.Google Scholar
Erdős, P. and Simonovits, M., Supersaturated graphs and hypergraphs , Combinatorica 3 (1983), 181–192; MR 726456.10.1007/BF02579292CrossRefGoogle Scholar
Fox, J., There exist graphs with super-exponential Ramsey multiplicity constant , J. Graph Theory 57 (2008), 89–98.10.1002/jgt.20256CrossRefGoogle Scholar
Fox, J. and Wei, F., On the local approach to Sidorenko’s conjecture, in The European conference on combinatorics, graph theory and applications (EUROCOMB’17), Electronic Notes in Discrete Mathematics, vol. 61, eds Drmota, M., Kang, M., Krattenthaler, C. and Nešetřil, J. (2017), 459–465.Google Scholar
Goodman, A. W., On sets of acquaintances and strangers at any party, Amer. Math. Monthly 66 (1959), 778–783.Google Scholar
Graham, R. L., Rothschild, B. L. and Spencer, J. H., Ramsey theory, Wiley Series in Discrete Mathematics and Optimization (John Wiley & Sons, Hoboken, NJ, 2013).Google Scholar
Grzesik, A., Lee, J., Lidický, B., and Volec, J., On tripartite common graphs, Combin. Probab. Comput. 1 (2022), 907–923.Google Scholar
Hancock, R., Král͏̌, D., Krnc, M. and Volec, J., Towards characterizing locally common graphs , Random Structures Algorithms 62 (2023), 181–218.10.1002/rsa.21099CrossRefGoogle Scholar
Hatami, H., Graph norms and Sidorenko’s conjecture, Israel J. Math. 175 (2010), 125–150.Google Scholar
Hatami, H., Hladký, J., D. Král͏̌, S. Norine and A. Razborov, Non-three-colourable common graphs exist, Combin. Probab. Comput. 21 (2012), 734–742.10.1017/S0963548312000107CrossRefGoogle Scholar
Hindman, N., Finite sums from sequences within cells of a partition of N, J. Combin. Theory Ser. A 17 (1974), 1–11; MR 349574.Google Scholar
Jagger, C., Šťovíček, P. and Thomason, A., Multiplicities of subgraphs , Combinatorica 16 (1996), 123–141.10.1007/BF01300130CrossRefGoogle Scholar
Ko, S. and Lee, J., Common graphs with arbitrary connectivity and chromatic number, J. Combin. Theory Ser. B 162 (2023), 223–230.10.1016/j.jctb.2023.06.001CrossRefGoogle Scholar
Kohayakawa, Y., Nagle, B., Rödl, V. and Schacht, M., Weak hypergraph regularity and linear hypergraphs, J. Combin. Theory Ser. B 100 (2010), 151–160.10.1016/j.jctb.2009.05.005CrossRefGoogle Scholar
Král͏̌, D., Noel, J. A., Norin, S., Volec, J. and Wei, F., Non-bipartite k-common graphs , Combinatorica 42 (2022), 87–114.10.1007/s00493-020-4499-9CrossRefGoogle Scholar
Lee, J., On some graph densities in locally dense graphs , Random Structures Algorithms 58 (2021), 322–344; MR 4201799.10.1002/rsa.20974CrossRefGoogle Scholar
Lovász, L., Subgraph densities in signed graphons and the local Simonovits-Sidorenko conjecture, Electron. J. Combin. 18 (2011), 127.10.37236/614CrossRefGoogle Scholar
Lovász, L., Large networks and graph limits, vol. 60 (AMS Colloquium Publications, 2012).10.1090/coll/060CrossRefGoogle Scholar
Ramsey, F. P., On a problem of formal logic, Proc. Lond. Math. Soc. (3) 2 (1930), 264–286.10.1112/plms/s2-30.1.264CrossRefGoogle Scholar
Razborov, A. A., Flag algebras, J. Symb. Logic 72 (2007), 1239–1282.10.2178/jsl/1203350785CrossRefGoogle Scholar
Reiher, C., Counting odd cycles in locally dense graphs, J. Combin. Theory Ser. B 105 (2014), 1–5.10.1016/j.jctb.2013.12.002CrossRefGoogle Scholar
Sah, A., Diagonal Ramsey via effective quasirandomness , Duke Math. J. 172 (2023), 545–567.10.1215/00127094-2022-0048CrossRefGoogle Scholar
Sidorenko, A., A correlation inequality for bipartite graphs, Graphs Combin. 9 (1993), 201–204.10.1007/BF02988307CrossRefGoogle Scholar
Sidorenko, A., Randomness friendly graphs , Random Structures Algorithms 8 (1996), 229–241.10.1002/(SICI)1098-2418(199605)8:3<229::AID-RSA6>3.0.CO;2-#3.0.CO;2-#>CrossRefGoogle Scholar
Sidorenko, A. F., Extremal problems in graph theory and functional-analytic inequalities, in Proc. of the All-Union Seminar on Discrete Mathematics and its Applications (Moskov. Gos. Univ., Mekh.-Mat. Fak., Moscow, 1986), 99–105 (in Russian); MR930303.Google Scholar
Sidorenko, A. F., Cycles in graphs and functional inequalities, Mat. Zametki 46 (1989), 72–79.Google Scholar
Sidorenko, A. F., Inequalities for functionals generated by bipartite graphs, Diskret. Mat. 3 (1991), 50–65.Google Scholar
Szegedy, B., An information theoretic approach to Sidorenko’s conjecture, Preprint (2014), arXiv:1406.6738.Google Scholar
Szemerédi, E., On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 199–245, collection of articles in memory of Juriĭ Vladimirovič Linnik.10.4064/aa-27-1-199-245CrossRefGoogle Scholar
Thomason, A., A disproof of a conjecture of Erdős in Ramsey theory, J. Lond. Math. Soc. (2) 39 (1989), 246–255.10.1112/jlms/s2-39.2.246CrossRefGoogle Scholar