Common graphs with arbitrary chromatic number

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, Erdos 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, Hladky, Kral, Norine and Razborov [Combin. Probab. Comput. 21 (2012), 734-742], and a problem listed by Conlon, Fox and Sudakov [London Math. Soc. Lecture Note Ser. 424 (2015), 49-118, Problem 2.28]. This also answers in a stronger form the question raised by Jagger, Stovicek and Thomason [Combinatorica 16, (1996), 123-131] whether there exists a common graph with chromatic number at least four.