Existence of common graphs with large chromatic number

Ramsey's Theorem states that for every graph H, there is an integer R(H) such that every 2-edge-coloring of R(H)-vertex complete graph contains a  monochromatic copy of H. In this talk, we focus on a natural quantitative extension: how many monochromatic copies of H can we find in every 2-edge-coloring of K_N, and, what graphs H are so-called common, i.e., the number of monochromatic copies of H  is asymptotically minimized by a random 2-edge-coloring.

A classical result of Goodman from 1959 states that triangle is a common graph. On the other hand, Thomason proved in 1989 that no clique of order at least four is common, and the existence of a common graph with chromatic number larger than three was open until about 10 years ago, when Hatami, Hladky, Kral, Norin and Razborov proved that the 5-wheel is common. In this talk, we show that for every k>4 there exists a common graph with chromatic number k.

This is a joint work with D. Kral and F. Wei