Erdős Problem 597

### What is the conjecture

https://www.erdosproblems.com/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 $G$?

Status: open

### Choose either option
- [ ] I plan on working on this conjecture
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else