SOLVED

Let $G$ be a graph on $n$ vertices with chromatic number $\chi(G)$ and let $\sigma(G)$ be the maximal $k$ such that $G$ contains a subdivision of $K_k$. Is it true that
\[\chi(G) \ll \frac{n^{1/2}}{\log n}\sigma(G)?\]

Hajós originally conjectured that $\chi(G)\leq \sigma(G)$, which was proved by Dirac [Di52] when $\chi(G)=4$. Catlin [Ca74] disproved Hajós' conjecture for all $\chi(G)\geq 7$, and Erdős and Fajtlowicz [ErFa81] disproved it in a strong form, showing that in fact for almost all graphs on $n$ vertices,
\[\chi(G) \gg \frac{n^{1/2}}{\log n}\sigma(G).\]

The answer is yes, proved by Fox, Lee, and Sudakov [FLS13].