Problem 597

Erdős Problem #597: declared status 'open'. Formalized: no. Let $G$ be a graph on at most $\aleph_1$ vertices which contains no $K_4$ and no $K_{\aleph_0,\aleph_0}$ (the complete bipartite graph with $\aleph_0$ vertices in each class). Is it true that$$\omega_1^2 \to (\omega_1\omega, G)^2?$$What about finite $G$? Prize: no. Tags: graph theory, ramsey theory, set theory.