Forcing-theoretic aspects of Hindman's Theorem

Abstract

We investigate the partial order $(\mathrm{FIN})^\omega$ of infinite block sequences, ordered by almost condensation, from the forcing-theoretic point of view. This order bears the same relationship to Hindman's Theorem as $\mathcal{P}(\omega) /\mathrm{fin}$ does to Ramsey's Theorem. While $(\mathcal{P}(\omega) / \mathrm{fin})^2$ completely embeds into $(\mathrm{FIN})^\omega$, we show this is consistently false for higher powers of $\mathcal{P} (\omega) / \mathrm{fin}$, by proving that the distributivity number $\mathfrak{h}_3$ of $(\mathcal{P} (\omega) /\mathrm{fin})^3$ may be strictly smaller than the distributivity number $\mathfrak{h}_{\mathrm{FIN}}$ of $(\mathrm{FIN})^\omega$. We also investigate infinite maximal antichains in $(\mathrm{FIN})^\omega$ and show that the least cardinality $\mathfrak{a}_{\mathrm{FIN}}$ of such a maximal antichain is at least the smallest size of a nonmeager set of reals. As a consequence, we obtain that $\mathfrak{a}_{\mathrm{FIN}}$ is consistently larger than $\mathfrak{a}$, the least cardinality of an infinite maximal antichain in $\mathcal{P} (\omega) / \mathrm{fin}$.