On the growth rate of chromatic numbers of finite subgraphs

We prove that, for every function $f:\mathbb{N}\to \mathbb{N}$, there is a graph $G$ with uncountable chromatic number such that, for every $k \in $\mathbb{N}$$with$k \geq 3$,everysubgraphof$G$withfewerthan$f(k)$verticeshaschromaticnumberlessthan$k$. This answers a question of Erd\H{o}s, Hajnal, and Szemeredi.