  For a graph $G$, let $\chi(G)$ denote its chromatic number and $\sigma(G)$
denote the order of the largest clique subdivision in $G$. Let H(n) be the
maximum of $\chi(G)/\sigma(G)$ over all $n$-vertex graphs $G$. A famous
conjecture of Haj\'os from 1961 states that $\sigma(G) \geq \chi(G)$ for every
graph $G$. That is, $H(n) \leq 1$ for all positive integers $n$. This
conjecture was disproved by Catlin in 1979. Erd\H{o}s and Fajtlowicz further
showed by considering a random graph that $H(n) \geq cn^{1/2}/\log n$ for some
absolute constant $c>0$. In 1981 they conjectured that this bound is tight up
to a constant factor in that there is some absolute constant $C$ such that
$\chi(G)/\sigma(G) \leq Cn^{1/2}/\log n$ for all $n$-vertex graphs $G$. In this
paper we prove the Erd\H{o}s-Fajtlowicz conjecture. The main ingredient in our
proof, which might be of independent interest, is an estimate on the order of
the largest clique subdivision which one can find in every graph on $n$
vertices with independence number $\alpha$.
