A weak variant of Hindman's Theorem stronger than Hilbert's Theorem
Abstract
Hirst investigated a slight variant of Hindman's Finite Sums Theorem  called Hilbert's Theorem  and proved it equivalent over $\RCA_0$ 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 variant of Hindman's Theorem  which we name the Adjacent Hindman's Theorem  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 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2016
 arXiv:
 arXiv:1610.05445
 Bibcode:
 2016arXiv161005445C
 Keywords:

 Mathematics  Logic;
 Mathematics  Combinatorics
 EPrint:
 Results from the literature imply stronger corollaries of results already proved in previous versions. Order of sections slightly changed