Restrictions of Hindman’s Theorem: An Overview

Carlucci, Lorenzo

doi:10.1007/978-3-030-80049-9_9

Cited by 3 publications

(3 citation statements)

References 16 publications

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“…In the usual Finite Unions versions of Hindman's Theorem the solution is a sequence (B i ) i∈N of finite subsets of the positive integers satisfying the so-called block condition: for all i < j, max(B i ) < min(B j ). When this condition is dropped the principle becomes much weaker, as proved by Hirst (see [5] references). We introduce the corresponding property in the finite sums setting.…”

Section: For Allmentioning

confidence: 90%

“…Restrictions of Hindman's Theorem relaxing the monochromaticity requirement to families of finite sums received substantial attention in recent years [5].…”

Section: Restrictions Of the Regressive Hindman's Theoremmentioning

confidence: 99%

“…Recently, substantial interest has been given to various restrictions of Hindman's Theorem (see [5] for an overview and references). Dzhafarov, Jockusch, Solomon and Westrick [10] proved that the only known lower bound on Hindman's Theorem already hits for Hindman's Theorem restricted to 3-colorings and sums of at most 3 terms (denoted HT ≤3 3 ) and that Hindman's Theorem restricted to 2-colorings and sums of at most 2 terms (denoted HT ≤2…”

Section: Introductionmentioning

confidence: 99%

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Regressive versions of Hindman's Theorem

Carlucci¹,

Mainardi²

2022

Preprint

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When the Canonical Ramsey's Theorem by Erdős and Rado is applied to regressive functions one obtains the Regressive Ramsey's Theorem by Kanamori and McAloon. Taylor proved a "canonical" version of Hindman's Theorem, analogous to the Canonical Ramsey's Theorem. We introduce the restriction of Taylor's Canonical Hindman's Theorem to a subclass of the regressive functions, the λ-regressive functions, relative to an adequate version of min-homogeneity and prove some results about the Reverse Mathematics of this Regressive Hindman's Theorem and of natural restrictions of it.In particular we prove that the first non-trivial restriction of the principle is equivalent to Arithmetical Comprehension. We furthermore prove that this same principle strongly computably reduces the well-ordering-preservation principle for base-ω exponentiation.

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Section: For Allmentioning

confidence: 90%

“…Restrictions of Hindman's Theorem relaxing the monochromaticity requirement to families of finite sums received substantial attention in recent years [5].…”

Section: Restrictions Of the Regressive Hindman's Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Regressive versions of Hindman's Theorem

Carlucci¹,

Mainardi²

2022

Preprint

Self Cite

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Regressive versions of Hindman’s theorem

Carlucci,

Mainardi

2024

Arch. Math. Logic

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When the Canonical Ramsey’s Theorem by Erdős and Rado is applied to regressive functions, one obtains the Regressive Ramsey’s Theorem by Kanamori and McAloon. Taylor proved a “canonical” version of Hindman’s Theorem, analogous to the Canonical Ramsey’s Theorem. We introduce the restriction of Taylor’s Canonical Hindman’s Theorem to a subclass of the regressive functions, the $$\lambda $$ λ -regressive functions, relative to an adequate version of min-homogeneity and prove some results about the Reverse Mathematics of this Regressive Hindman’s Theorem and of natural restrictions of it. In particular we prove that the first non-trivial restriction of the principle is equivalent to Arithmetical Comprehension. We furthermore prove that the well-ordering-preservation principle for base-$$\omega $$ ω exponentiation is reducible to this same principle by a uniform computable reduction.

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A RECURSIVE COLORING FUNCTION WITHOUT $ \Pi _3^0$ SOLUTIONS FOR HINDMAN’S THEOREM

LIAO

2024

J. symb. log.

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Restrictions of Hindman’s Theorem: An Overview

Cited by 3 publications

References 16 publications

Regressive versions of Hindman's Theorem

Regressive versions of Hindman's Theorem

Regressive versions of Hindman’s theorem

A RECURSIVE COLORING FUNCTION WITHOUT $ \Pi _3^0$ SOLUTIONS FOR HINDMAN’S THEOREM

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