Thin Set Versions of Hindman's Theorem

TL;DR

This paper investigates a variation of Hindman's Theorem called thin-HT, analyzing its logical strength and computational complexity, and establishing its implications for reverse mathematics and computability theory.

Contribution

It introduces thin-HT, explores its reverse mathematical strength, and demonstrates its computational complexity properties, including implications for DNC degrees and the arithmetical hierarchy.

Findings

thin-HT implies ACA_0 over RCA_0 + IΣ^0_2

computable instances of thin-HT have solutions computing ∅'

restricted thin-HT solutions can be diagonally noncomputable relative to ∅'

Abstract

In this paper we examine the reverse mathematical strength of a variation of Hindman's Theorem HT constructed by essentially combining HT with the Thin Set Theorem TS to obtain a principle which we call thin-HT. thin-HT says 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 $c(s) \neq i$ for all $s \in S$. We show that there is a computable instance of thin-HT such that every solution computes $\emptyset'$, as is the case with HT (see Blass, Hirst, and Simpson 1987). In analyzing this proof, we deduce that thin-HT implies $ACA_0$ over $RCA_0 + I\Sigma^0_2$. On the other hand, using Rumyantsev and Shen's computable version of the Lov\'asz 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 $\emptyset'$. Hence there is a computable instance of this restriction of thin-HT with no $\Sigma^0_2$ solution.