Claims

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1971_09_01_erdos_hajnal: Erdős and Hajnal (Period. Math. Hungar., 1971) proved in ZFC that omega_1^2 -> (omega_1 omega, 3)^2, which answers the finite question for the triangle and so for every graph on at most three vertices.

1989_01_01_baumgartner: Baumgartner (Lecture Notes in Math. 1401, 1989) proved under Martin's axiom for aleph_1 dense sets that omega_1 omega -> (omega_1 omega, n)^2 for all finite n; so the finite question holds in a model of ZFC for every graph.

2026_07_27_white: Claims in ZFC the relation for every finite K_4-free block graph and reduces the finite question to the 2-connected K_4-free graphs, with C_4 and K_4-e as the first cases left open.

2026_09_09_li: Claims, granted Baumgartner's relation, a bipartite target on aleph_1 vertices with no complete bipartite subgraph whose two sides are countably infinite for which the relation fails, answering the first question negatively.

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