Archive for Mathematical Logic

, Volume 57, Issue 3–4, pp 381–389 | Cite as

A weak variant of Hindman’s Theorem stronger than Hilbert’s Theorem

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Abstract

Hirst investigated a natural restriction of Hindman’s Finite Sums Theorem—called Hilbert’s Theorem—and proved it equivalent over \(\mathbf {RCA}_0\) to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman’s Theorem provably much weaker than Hindman’s Theorem itself. We here introduce another natural restriction of Hindman’s Theorem—which we name the Adjacent Hindman’s Theorem with apartness—and prove it to be provable from Ramsey’s Theorem for pairs and strictly stronger than Hirst’s Hilbert’s Theorem. The lower bound is obtained by a direct combinatorial implication from the Adjacent Hindman’s Theorem with apartness to the Increasing Polarized Ramsey’s Theorem for pairs introduced by Dzhafarov and Hirst. In the Adjacent Hindman’s Theorem homogeneity is required only for finite sums of adjacent elements.

Keywords

Hindman’s Theorem Computable combinatorics Ramsey’s Theorem Reverse mathematics 

Mathematics Subject Classification

05D10 03B30 03F35 

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Dipartimento di InformaticaSapienza — Università di RomaRomeItaly

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