A problem of Erd\H{o}s, Hajnal, and Milner \cite{EHM70}, who proved this is true for $\alpha < \omega_1^{\omega+2}$.

In \cite{Er82e} Erd\H{o}s offers \$250 for showing what happens when $\alpha=\omega_1^{\omega+2}$ and \$500 for settling the general case.

Larson \cite{La90} proved this is true for all $\alpha<2^{\aleph_0}$ assuming Martin's axiom.


References


[EHM70] Erd\H{o}s, P. and Hajnal, A. and Milner, E. C., Set mappings and polarized partition relations. Combinatorial theory and its applications, I-III (Proc.
Colloq., Balatonf\"{u}red, 1969) (1970), 327-363.

[Er82e] Erd\H{o}s, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79.

[La90] Larson, Jean A., Martin's axiom and ordinal graphs: large independent sets or infinite paths. Ann. Pure Appl. Logic (1990), 31-39.