OPEN

Let $\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\mathfrak{m}$. Let $C\geq 1$. Must $G$ contain a subgraph of chromatic number $\mathfrak{m}$ which does not contain any odd cycle of length $\leq C$?

A question of Erdős and Hajnal. Rödl proved this is true if $\mathfrak{m}=\aleph_0$ and $C=3$.

In [Er81] Erdős also asks the same question but with girth (i.e. the subgraph does not contain any cycle at all of length $\leq C$).