Notre Dame Journal of Formal Logic 53 (3):373-395 (2012)

Abstract
We study the reverse mathematics of pigeonhole principles for finite powers of the ordinal $\omega$ . Four natural formulations are presented, and their relative strengths are compared. In the analysis of the pigeonhole principle for $\omega^{2}$ , we uncover two weak variants of Ramsey’s theorem for pairs
Keywords reverse mathematics   indecomposability   partition principles
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DOI 10.1215/00294527-1716784
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On the Strength of Ramsey's Theorem.David Seetapun & Theodore A. Slaman - 1995 - Notre Dame Journal of Formal Logic 36 (4):570-582.
A Cohesive Set Which is Not High.Carl Jockusch & Frank Stephan - 1993 - Mathematical Logic Quarterly 39 (1):515-530.
The Polarized Ramsey’s Theorem.Damir D. Dzhafarov & Jeffry L. Hirst - 2009 - Archive for Mathematical Logic 48 (2):141-157.

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