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

Hirst investigated a slight variant of Hindman's Finite Sums Theorem called Hilbert's Theorem and proved it equivalent over RCA0 to the In nite Pigeonhole Principle for all colors. This gave the rst 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 nite sums of adjacent elements.