We study the reverse mathematics of pigeonhole principles for finite powers of the ordinal . Four natural formulations are presented, and their relative strengths are compared. In the analysis of the pigeonhole principle for , we uncover two weak variants of Ramsey’s theorem for pairs.
Jared R. Corduan. François G. Dorais. "On the Indecomposability of ." Notre Dame J. Formal Logic 53 (3) 373 - 395, 2012. https://doi.org/10.1215/00294527-1716784