Infinite graphs have infinite paths or arbitrary independent sets

A problem on ordinal graphs and infinite paths (proposed by Erdös, Hajnal and Milner [1])
For which limit ordinals \( \alpha\) is it true that if \( G\) is a graph whose vertices form a set of order type \( \alpha\), then either \( G\) has an infinite path or contains an independent set of order type \( \alpha\). In other words, determine the limit ordinals \( \alpha\) for which

\(\displaystyle \alpha\rightarrow(\alpha,\)infinite  path\(\displaystyle )^2. \)

Erdös, Hajnal and Milner [1] proved that the positive relation is true for all limit \( \alpha<\omega^{\omega+2}\). Baumgartner and Larson [2] showed that if Jensen's Diamond Principle holds, then \( \alpha\not\rightarrow(\alpha,\)infinite path\( )^2\) for all \( \alpha\) with \( \omega_1^{\omega+2}\le \alpha<\omega_2\). Larson [3] obtained further results under the assumption of GCH.


Bibliography
1 P. Erdös, A. Hajnal and E. Milner, Set mappings and polarized partition relations, Combinatorial theory and its applications, I (Proc. Colloq., Balatonfüred, 1969), pp. 327-363, North-Holland, Amsterdam, 1970.
2 J. E. Baumgartner and J. A. Larson, A Diamond example of an ordinal graph with no infinite paths, Annals of Pure and Appl. Logic, 47 (1990), 1-10.
3 J. A. Larson, A GCH example of an ordinal graph with no infinite path, Trans. Amer. Math. Soc. 303 (1987), 383-393.