New bounds on the strength of some restrictions of Hindman’s Theorem

Abstract

The relations between (restrictions of) Hindman's Finite Sums Theorem and (variants of)Ramsey's Theorem give rise to long-standing open problems in combinatorics, computability theory and proof theory. We present some results motivated by these open problems. In particular we investigate the restriction of the Finite Sums Theorem to sums of one or two elements, which is the subject of a long-standing open question by Hindman, Leader and Strauss. We show that this restriction has the same proof-theoretical and computabilitytheoretic lower bound that is known to hold for the full version of the Finite Sums Theorem. In terms of reverse mathematics, it implies ACA 0 . Also, we show that Hindman's Theorem restricted to sums of exactly n ≥ 3 elements, is equivalent to ACA 0 , provided a certain sparsity condition is imposed on the solution set. The same results apply to bounded versions of the Finite Union Theorem, in which such a sparsity condition is built-in. Further we show that the Finite Sums Theorem for sums of at most two elements is tightly connected to the Increasing Polarized Ramsey's Theorem for pairs introduced by Dzhafarov and Hirst. The latter reduces to the former in a strong technical sense known as strong computable reducibility, which essentially means that there is a natural combinatorial reduction proof of one principle to the other.

The reverse mathematics of Hindman’s Theorem for sums of exactly two elements

“…Let M be as in Corollary 2.3 for q = 1 2 . We may assume that M > 0 and M is sufficiently large so that bm 2…”

“…We thank Jason Bell and Jeff Shallit for very helpful suggestions that led us to the idea of using the Lovász Local Lemma. We also thank Ludovic Patey for information on RRT 2 2 .…”

“…Dzhafarov, Jockusch, Solomon, and Westrick [4] showed that HT 3 3 implies ACA 0 over RCA 0 and that HT 2 2 is not provable in RCA 0 . Carlucci, Ko lodzieczyk, Lepore, and Zdanowski [2] investigated versions of Hindman's Theorem for sums of bounded length in which the solutions are required to meet a certain natural sparseness condition known as apartness, and in particular showed that HT 2 2 with apartness implies ACA 0 over RCA 0 . (They also have results for HT =n k with apartness.)…”