On the indecomposability of $\omega^n$
  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. 
Jared R. Corduan, Fran\c{c}ois G. Dorais
math.LO