A forcing notion related to Hindman's theorem (Q2257109)

\textit{N. Hindman}'s finite sum theorem [J. Comb. Theory, Ser. A 17, 1--11 (1974; Zbl 0285.05012)] states that, given a coloring of the natural numbers \(\mathbb N\) into finitely many colours, there is an infinite set \(A\subseteq \mathbb N\) such that all sums of finitely many elements of \(A\) (no repetition allowed) have the same color. Ever since \textit{A. R. D. Mathias}' analysis of selective co-ideals a.k.a. happy families [Ann. Math. Logic 12, 59--111 (1977; Zbl 0369.02041)], a connection between Ramsey-type statements and combinatorics of related forcing notions has been studied (see [\textit{S. Todorcevic}, Introduction to Ramsey space. Princeton, NJ: Princeton University Press (2010; Zbl 1205.05001)] for an excellent general treatment of the subject). The paper under review contributes to this line of research by introducing and studying a natural partial order, denoted by \(\mathbb P_{\mathrm{FIN}}\), corresponding to Hindman's theorem much in the same way as Mathias forcing relates to the infinite version of the Ramsey theorem. The main results of the paper show that this partial order is indeed new in that it is not forcing equivalent to the similar, previously introduced, partial orders due to Mathias [loc. cit.] and \textit{P. Matet} [J. Symb. Log. 53, No. 2, 540--553 (1988; Zbl 0655.04003)].