Thin Set Versions of Hindman’s Theorem

Abstract

We examine the reverse mathematical strength of a variation of Hindman’s Theorem (HT) constructed by essentially combining HT with the Thin Set Theorem to obtain a principle that we call thin-HT. This principle states that every coloring $c:\mathbb{N}\to \mathbb{N}$ has an infinite set $S\subseteq \mathbb{N}$ whose finite sums are thin for c, meaning that there is an i with $c(s)\ne i$ for all nonempty sums s of finitely many distinct elements of S. We show that there is a computable instance of thin-HT such that every solution computes ${\mathrm{\varnothing }}^{\prime }$, as is the case with HT, as shown by Blass, Hirst, and Simpson (1987). In analyzing this proof, we deduce that thin-HT implies ${\mathrm{ACA}}_0$ over ${\mathrm{RCA}}_0+\mathrm{I}{\mathrm{\Sigma }}_2^0$. On the other hand, using Rumyantsev and Shen’s computable version of the Lovász Local Lemma, we show that there is a computable instance of the restriction of thin-HT to sums of exactly 2 elements such that any solution has diagonally noncomputable degree relative to ${\mathrm{\varnothing }}^{\prime }$. Hence there is a computable instance of this restriction of thin-HT with no ${\mathrm{\Sigma }}_2^0$ solution.