Thin Set Versions of Hindman’s Theorem

Abstract

This paper is part of a line of research on the computability-theoretic and reverse-mathematical strength of versions of Hindman’s Theorem [6] that began with the work of Blass, Hirst, and Simpson [1], and has seen considerable interest recently. We assume basic familiarity with computability theory and reverse mathematics, at the level of the background material in [8], for instance. On the reverse mathematics side, the two major systems with which we will be concerned are RCA0, the usual weak base system for reverse mathematics, which corresponds roughly to computable mathematics; and ACA0, which corresponds roughly to arithmetic mathematics. For principles P of the form (∀X) [Φ(X) → (∃Y ) Ψ(X, Y )], we call any X such that Φ(X) holds an instance of P , and any Y such that Ψ(X, Y ) holds a solution to X . We begin by introducing some related combinatorial principles. For a set S, let [S] be the set of n-element subsets of S. Ramsey’s Theorem (RT) is the statement that for every n and every coloring of [N] with finitely many colors, there is an infinite set H that is homogeneous for c, which means that all elements of [H ] have the same color. There has been a great deal of work on computability-theoretic and reverse-mathematical aspects of versions of Ramsey’s Theorem, such as RTnk , which is RT restricted to colorings of [N] n with k many colors. (See e.g. [8].)