Schipperus, Rene, Countable partition ordinals. Ann. Pure Appl. Logic 161 (2010), 1195--1215, doi:10.1016/j.apal.2009.12.007 (received 9 May 2007, accepted 26 December 2009, available online 13 May 2010, per p. 1195). Theorem 28, p. 1212, "Let $\beta<\omega_1$ be the sum of one or two indecomposable ordinals, then $\omega^{\omega^\beta}\to(\omega^{\omega^\beta},3)^2$", whose case $\beta=2=1+1$ is this problem's relation, written out on pp. 1197 and 1215; the statement, in its three printed forms (Theorem 1, p. 1195; Theorem 3, p. 1196; Theorem 28), and the one-paragraph proof of Theorem 28 are the basis, the supporting Sections 2--10 for structure only. Library home: Schipperus 2010 and its Schipperus 2010, Theorem 28 page.

Specker, Ernst, Teilmengen von Mengen mit Relationen. Comment. Math. Helv. (1957), 302-314.