Problem 1091

Erdős Problem #1091: declared status 'solved'. Formalized: no. Let $G$ be a $K_4$-free graph with chromatic number $4$. Must $G$ contain an odd cycle with at least two diagonals? More generally, is there some $f(r)\to \infty$ such that every graph with chromatic number $4$, in which every subgraph on $\leq r$ vertices has chromatic number $\leq 3$, contains an odd cycle with at least $f(r)$ diagonals? Prize: no. Tags: chromatic number, geometry.