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 at most 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-theoretic and computability-theoretic 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 elements is equivalent to ACA 0 for each n 3, 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 already 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 the technical sense known as strong computable reducibility.

Thin Set Versions of Hindman's Theorem

“…Dzhafarov, Jockusch, Solomon, and Westrick [5] showed that HT 3 3 implies ACA 0 over RCA 0 . Carlucci, Ko lodzieczyk, Lepore, and Zdanowski [3] did the same for HT 2 4 . These principles are also complex in a more heuristic sense: There is no known way to prove even HT 2 2 other than to give a proof of the full HT, which has led Hindman, Leader, and Strauss [7] to ask whether every proof of HT 2 is also a proof of HT.…”

“…One might hope to show, for instance, that there is a boundary between principles that "behave like HT", e.g. HT 2 4 , which as mentioned in the introduction was shown to imply ACA 0 in [3]; and those that "behave like versions of TS / RT", e.g. the thin version of HT 2 4 , which can easily be shown to follow from RT 2 4,2 .…”

“…Jockusch [9] showed that there is a computable instance of RT 3 2 such that any solution computes the halting problem ∅ ′ . As shown by Simpson [18], Jockusch's construction can also be used to prove that RT 3 2 (and hence RT) implies ACA 0 over RCA 0 . Wang [19] showed that TS, on the other hand, does not have this much power.…”

Logic Blog 2017

“…(1) For every positive integers k, ℓ, RCA 0 ⊢ HT ≤3 3 → RT k ℓ (Dzhafarov et al [7]), (2) For every positive integers k, ℓ, RCA 0 ⊢ HT ≤2 4 → RT k ℓ (Carlucci et al [4]), (3) For every positive integers k, ℓ, RCA 0 ⊢ HT =3 2 with apartness → RT k ℓ (Carlucci et al [4]), (4) IPT 2 2 ≤ sc HT ≤2 4 (Carlucci [3]), (5) IPT 2 2 ≤ sc HT =2 2 with apartness (Carlucci et al [4]). Despite points (1) and (2), I have not been able to lift the combinatorial reductions in points ( 3) and ( 4) to exponents higher than 2.…”

Regressive versions of Hindman's Theorem

“…2 ) is unprovable in RCA 0 . The first author in joint work with Ko lodziejczyk, Lepore and Zdanowski later showed that HT ≤2 4 implies ACA 0 and that HT ≤2 2 is unprovable in WKL 0 [4]. However, no upper bounds other than those known for the full Hindman's Theorem are known for HT ≤2 k , let alone HT ≤3 k , for any k > 1.…”