Effectiveness of Hindman’s Theorem for Bounded Sums

Abstract

We consider the strength and effective content of restricted versions of Hindman’s Theorem in which the number of colors is specified and the length of the sums has a specified finite bound. Let $\mathsf {HT}^{\le n}_k$ denote the assertion that for each k-coloring c of $\mathbb {N}$ there is an infinite set $X \subseteq \mathbb {N}$ such that all sums $\sum _{x \in F} x$ for $F \subseteq X$ and $0 < |F| \le n$ have the same color. We prove that there is a computable 2-coloring c of $\mathbb {N}$ such that there is no infinite computable set X such that all nonempty sums of at most 2 elements of X have the same color. It follows that $\mathsf {HT}^{\le 2}_2$ is not provable in $\mathsf {RCA}_0$ and in fact we show that it implies $\mathsf {SRT}^2_2$ in $\mathsf {RCA}_0+ \mathsf {B}\Pi ^0_1$. We also show that there is a computable instance of $\mathsf {HT}^{\le 3}_3$ with all solutions computing $0'$. The proof of this result shows that $\mathsf {HT}^{\le 3}_3$ implies $\mathsf {ACA}_0$ in $\mathsf {RCA}_0$.

This paper is dedicated to Rod Downey in honor of his outstanding contributions to computability theory and his leadership role in mentoring and exposition. Two of the authors had the pleasure of being mentored by Rod as postdocs.

Dzhafarov was partially supported by NSF grant DMS-1400267. Jockusch thanks his coauthors for their hospitality during a visit to the University of Connecticut in May, 2015, during which the work in this paper was largely done. His visit was supported by the University of Connecticut Department of Mathematics. The authors thank Denis Hirschfeldt for numerous valuable insights.