  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.
