  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.
