Effectiveness of Hindman’s Theorem for Bounded Sums

Dzhafarov, Damir D.; Jockusch, Carl G.; Westrick, Linda Brown

doi:10.1007/978-3-319-50062-1_11

Cited by 11 publications

(30 citation statements)

References 14 publications

(15 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section we first show that HT ≤2 implies ACA 0 (hence RT 3 2 ) over RCA 0 . This improves on the main result of [15] that HT ≤3 implies ACA 0 . In particular we show that HT ≤2 4 implies ACA 0 .…”

Section: Bounded Hindman Vs Ramseysupporting

confidence: 74%

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

Carlucci

Kołodziejczyk

Lepore

et al. 2020

COM

View full text Add to dashboard Cite

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.

show abstract

Section: Bounded Hindman Vs Ramseysupporting

confidence: 74%

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

Carlucci

Kołodziejczyk

Lepore

et al. 2020

COM

View full text Add to dashboard Cite

show abstract

“…Recently there has been some interest in the computability-theoretic and proof-theoretic strength of restrictions of Hindman's Theorem (see [10,5,4]). While [10] deals with a restriction on the sequence of finite sets in the Finite Unions formulation of Hindman's Theorem, both [5] and [4] deal with restrictions on the types of sums that are guaranteed to be colored the same color.…”

Section: Introductionmentioning

confidence: 99%

“…Let us denote by HT ≤n r the restriction of the Finite Sums Theorem to colorings with r colors and sums of at most n terms. The conjecture discussed in [1] is then that the complexity of HT ≤n r is growing with n. The main result in [5] is that the above described ∅ ′ lower bound known to hold for the full Hindman's Theorem already applies to its restriction to 4 colors and to sums of at most 3 terms (HT ≤3 4 in the notation introduced above). On the other hand, note that no upper bound other than the upper bound for full Hindman's Theorem is known to hold for this restricted version, and the same is true for HT ≤2 2 , the restriction to sums of at most 2 terms!…”

Section: Introductionmentioning

confidence: 99%

“Weak yet strong” restrictions of Hindman’s Finite Sums Theorem

Carlucci

2017

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

We present a natural restriction of Hindman's Finite Sums Theorem that admits a simple combinatorial proof (one that does not also prove the full Finite Sums Theorem) and low computability-theoretic and proof-theoretic upper bounds, yet implies the existence of the Turing Jump, thus realizing the only known lower bound for the full Finite Sums Theorem. This is the first example of this kind. In fact we isolate a rich family of similar restrictions of Hindman's Theorem with analogous properties.

show abstract

“…We first show that it is very easy to establish an upper bound on AHT 2 and AHT. This should be contrasted with the case of Hindman's Theorem restricted to sums of at most two terms (HT ≤2 2 in the notation of [7]), for which no upper bound other than ACA + 0 is currently known.…”

Section: Upper Bound: Adjacent Hindman's Theorem Follows From Ramsey mentioning

confidence: 99%

“…Recently there has been some interest in the strength of restrictions of Hindman's Theorem (see [9,7]). Interestingly, Dzhafarov, Jockusch, Solomon and Westrick [7] proved that the only known lower bound on Hindman's Theorem already hits for HT ≤3 4 (Hindman's Theorem restricted to 4-colorings and sums of at most 3 terms) and that HT ≤2 2 (Hindman's Theorem restricted to 2-colorings and sums of at most 2 terms) is unprovable in RCA 0 ). However, no upper bounds other than those known for the full Hindman's Theorem are known for HT ≤2 2 , let alone HT ≤3 4 .…”

Section: Introductionmentioning

confidence: 99%