  We prove that, for every function $f:\mathbb{N} \rightarrow \mathbb{N}$,
there is a graph $G$ with uncountable chromatic number such that, for every $k
\in \mathbb{N}$ with $k \geq 3$, every subgraph of $G$ with fewer than $f(k)$
vertices has chromatic number less than $k$. This answers a question of
Erd\H{o}s, Hajnal, and Szemeredi.
