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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
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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
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence.

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