Forcing-theoretic aspects of Hindman's theorem (Q2407561)

Let \(\mathrm{FIN}\) denote the family of non-empty finite subsets of \(\omega\). A sequence \(D=(d_i:i\in\omega)\) of elements of \(\mathrm{FIN}\) is called a block sequence if \(\max d_i<\min d_{i+1}\) for all \(i\in\omega\). Let \((\mathrm{FIN})^\omega\) denote the collection of infinite block sequences and for \(D\in(\mathrm{FIN})^\omega\) let \(\mathrm{FU}(D)\) be the set of all nonempty finite unions of elements of \(D\). If \(E=(e_i:i\in\omega)\in(\mathrm{FIN})^\omega\) then we write \(E\sqsubseteq^*D\) if there is \(n\in\omega\) such that \(e_i\in\mathrm{FU}(D)\) for all \(i\geq n\). It is easy to see that \(((\mathrm{FIN})^\omega,{\sqsubseteq^*})\) is a \(\sigma\)-closed forcing notion. This forcing notion generically adds an ultrafilter \(\mathcal{U}\) such that witnesses for Hindman's theorem can be found within \(\mathcal{U}\). The ultrafilter \(\mathcal{U}\) is moreover stable which means that if \(D_n\in(\mathrm{FIN})^\omega\) are such that \(\mathrm{FU}(D_n)\in\mathcal{U}\) and \(D_{n+1}\sqsubseteq^*D_n\) for all \(n\in\omega\), then there is \(E\in(\mathrm{FIN})^\omega\) such that \(\mathrm{FU}(E)\in\mathcal{U}\) and \(E\sqsubseteq^*D_n\) for all \(n\in\omega\). Consider the distributivity numbers \(\mathfrak{h}_{\mathrm{FIN}}=\mathfrak{h}((\mathrm{FIN})^\omega)\) and \(\mathfrak{h}_n=\mathfrak{h}((\mathcal{P}(\omega)/\mathrm{fin})^n)\) for \(n\in\omega\) and the almost disjointness number \(\mathfrak{a}_{\mathrm{FIN}}=\mathfrak{a}((\mathrm{FIN})^\omega)\). Under CH, all \(\sigma\)-closed forcing notions are forcing equivalent to \(\mathcal{P}(\omega)/\mathrm{fin}\). Hence, the real question for such forcing notions is whether a complete embedability is provable in ZFC or whether it consistently fails. The authors show that \((\mathcal{P}(\omega)/\mathrm{fin})^2\) completely embeds into \((\mathrm{FIN})^\omega\) but this is consistently false for higher powers of \(\mathcal{P}(\omega)/\mathrm{fin}\) because \(\mathfrak{h}_3\) may be strictly smaller than \(\mathfrak{h}_{\mathrm{FIN}}\). On the other hand, they prove that it is consistent that \(\mathfrak{h}_{\mathrm{FIN}}<\mathfrak{h}_n\) for all \(n\). The authors also investigate maximal antichains in \((\mathrm{FIN})^\omega\) and prove that \(\mathfrak{a}_{\mathrm{FIN}}\) is above the minimal size of a nonmeager set of reals and each of the inequalities \(\mathfrak{a}<\mathfrak{a}_{\mathrm{FIN}}\) and \(\mathfrak{a}_{\mathrm{FIN}}<\mathfrak{c}\) is consistent.