Chang, C. C., A partition theorem for the complete graph on $\omega\sp{\omega }$. J. Combinatorial Theory Ser. A 12 (1972), 396--452; doi:10.1016/0097-3165(72)90105-7 (received 24 February 1970; the running head prints volume 12). This problem's question in the form "$\omega^{\omega^\alpha}\to(\omega^{\omega^\alpha},3)^2$ if $\alpha<\omega_1$?", posed as one of two representative unknown problems, p. 397; the Theorem $\omega^\omega\to(\omega^\omega,3)^2$, the case $\alpha=1$ in the paper's notation $\omega^{\omega^\alpha}$ (the problem's case $\beta=\omega$), p. 396; footnote 1 with Milner's $\omega^\omega\to(\omega^\omega,m)^2$, $m<\omega$, reported by letter, p. 397; all cited at statement depth. Library home: Chang 1972 and its Chang 1972, Problems (p. 397) and Chang 1972, Theorem (p. 396) pages.

Galvin, Fred and Larson, Jean, Pinning countable ordinals. Fund. Math. 82 (1974/75), 357-361.

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). The question in the paper's form, p. 1196 ("for which countable $\beta$ does $\omega^{\omega^\beta}\to(\omega^{\omega^\beta},3)^2$?", after the Galvin--Larson reduction [GaLa74] to $\omega^2$ and the ordinals $\omega^{\omega^\beta}$; the paper's $\beta$ is the problem's $\gamma$, with $\alpha=\omega^{\omega^\gamma}$); Theorem 28, p. 1212, yes for the paper's $\beta$ the sum of one or two indecomposable ordinals; Theorem 29, p. 1213 (Theorems 31--33, pp. 1214--1215), $\not\to(\omega^{\omega^\beta},6)^2$ for two indecomposables, $\not\to(\omega^{\omega^\beta},4)^2$ for three and $\not\to(\omega^{\omega^\beta},3)^2$ for four or more, which leaves the 3-relation for the sum of three indecomposables undecided; all cited at statement depth. Library home: Schipperus 2010 and its Schipperus 2010, Theorem 28 and Schipperus 2010, Theorem 29 pages.

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