Forcing-theoretic aspects of Hindman's Theorem

Abstract

We investigate the partial order (FIN)^ω 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 $\mathcal{P}$(ω)/fin does to Ramsey's Theorem. While ($\mathcal{P}$(ω)/fin)² completely embeds into (FIN)^ω, we show this is consistently false for higher powers of $\mathcal{P}$(ω)/fin, by proving that the distributivity number 𝔥₃ of ($\mathcal{P}$(ω)/fin)³ may be strictly smaller than the distributivity number 𝔥_FIN of (FIN)^ω. We also investigate infinite maximal antichains in (FIN)^ω and show that the least cardinality 𝔞_FIN of such a maximal antichain is at least the smallest size of a nonmeager set of reals. As a consequence, we obtain that 𝔞_FIN is consistently larger than 𝔞, the least cardinality of an infinite maximal antichain in $\mathcal{P}$(ω)/fin.