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

Lorenzo Carlucci

Published 2016-10-18Version 1

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. This gave the first example of a finitistically reducible variant of Hindman's Theorem. We here introduce another natural variant of Hindman's Theorem -- which we name the Adjacent Hindman's Theorem -- and prove it to be strictly stronger than the Infinite Pigeonhole Principle and at most as strong as Ramsey's Theorem for colorings of pairs in $2$ colors (in the sense of Reverse Mathematics). In the Adjacent Hindman's Theorem homogeneity is required only for finite sums of adjacent elements.