On the growth rate of chromatic numbers of finite subgraphs

Abstract

We prove that, for every function f:N→N, there is a graph G with uncountable chromatic number such that, for every k≥3, every subgraph of G with fewer than f(k) vertices has chromatic number less than k. This answers a question of Erdős, Hajnal, and Szemerédi. We also show that, if ⋄ holds, then we can take this graph G to be a Hajnal-Máté graph on ω1.