OPEN

Let $\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\mathfrak{m}$. Is it true that, for every infinite cardinal $\mathfrak{n}< \mathfrak{m}$, there exists a subgraph of $G$ with chromatic number $\mathfrak{n}$?

A question of Galvin, who proved that the answer is no if we ask for the subgraph to be induced (assuming $\aleph_1 < 2^{\aleph_0}$).