Darby, Carl, Negative partition relations for ordinals $\omega^{\omega^\alpha}$. J. Combin. Theory Ser. B (1999), 205-222.

Foreman, Matthew and Kanamori, Akihiro, Handbook of set theory. Vols. 1, 2, 3. (2010), Vol. 1: xiv+736 pp.; Vol. 2: pp. i-xiv and 737-1447; Vol. 3: pp. i-xiv and 1449-2197.

Larson, Jean A., An ordinal partition avoiding pentagrams. J. Symbolic Logic (2000), 969-978.

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). Its closing remark, p. 1215, "$\omega^{\omega^2}\to(\omega^{\omega^2},3)^2$ but $\omega^{\omega^2}\not\to(\omega^{\omega^2},6)^2$. Thus it is not true that $\alpha\to(\alpha,3)^2$ implies $\alpha\to(\alpha,n)^2$ for all $n<\omega$", follows from Theorem 28, p. 1212, and Theorem 29(1), p. 1213 (proved as Theorem 31, p. 1214); the abstract announces the example, and p. 1197 attributes the question to Specker (1957) and Erdős (1992) and reports Larson's sharpening of the 6 to a 5 [La00]. Library home: Schipperus 2010 and its Schipperus 2010, Theorem 28 and Schipperus 2010, Theorem 29 pages.

Schipperus, Rene J., Countable partition ordinals. (1999), 57.