Erdős Problem #919 — Is there a graph G with vertex set omega_2^2 and chromatic number aleph_2 such that every subgraph whose vertices have a lesser type has chromatic number ≤ aleph_0?

Question

Is there a graph G with vertex set omega_2^2 and chromatic number aleph_2 such that every subgraph whose vertices have a lesser type has chromatic number ≤ aleph_0? What if instead we ask for G to have chromatic number aleph_1?

Erdős Problem #919 — Is there a graph $G$ with vertex set $\omega_2^2$ and chromatic number $\aleph_2$ such that every subgraph whose vertices have a lesser type has chromatic number $\leq \aleph_0$ ?

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