Problem 597

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 $G$ ?

Source: erdosproblems.com/597

Graph theory
Ramsey theory
Set theory

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Problem 597

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