Regressive versions of Hindman's Theorem

TL;DR

This paper introduces a restricted version of Taylor's canonical Hindman's Theorem focusing on λ-regressive functions, exploring its logical strength and computational properties within Reverse Mathematics.

Contribution

It defines λ-regressive functions and establishes the logical strength of their associated Hindman's Theorem, linking it to Arithmetical Comprehension and well-ordering principles.

Findings

First non-trivial restriction equivalent to Arithmetical Comprehension

Strongly reduces the well-ordering-preservation principle for base-ω exponentiation

Advances understanding of the reverse mathematical strength of canonical Hindman's Theorem

Abstract

When the Canonical Ramsey's Theorem by Erd\H{o}s 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 $\lambda$-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-$\omega$ exponentiation.