OPEN

Let $G$ be a graph with chromatic number $\aleph_1$. Is there, for any integer $m\geq 1$, some graph $G_m$ of chromatic number $m$ such that every finite subgraph of $G_m$ is a subgraph of $G$?

A conjecture of Walter Taylor.

More generally, Erdős asks to characterise families $\mathcal{F}_\alpha$ of finite graphs such that there is a graph of chromatic number $\aleph_\alpha$ all of whose finite subgraphs are in $\mathcal{F}_\alpha$.