Hindman’s theorem for sums along the full binary tree, \(\Sigma ^0_2\)-induction and the Pigeonhole principle for trees

Abstract

We formulate a restriction of Hindman’s Finite Sums Theorem in which monochromaticity is required only for sums corresponding to rooted finite paths in the full binary tree. We show that the resulting principle is equivalent to \(\Sigma ^0_2\)-induction over \(\mathsf {RCA}_0\). The proof uses the equivalence of this Hindman-type theorem with the Pigeonhole Principle for trees \({\mathsf {T}\,}{\mathsf {T}}^1\) with an extra condition on the solution tree.