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 the full binary tree. show that resulting principle equivalent $$\Sigma ^0_2$$ Σ 2 0 -induction over $$\mathsf {RCA}_0$$ RCA . The proof uses equivalence this Hindman-type theorem with Pigeonhole Principle trees $${\mathsf {T}\,}{\mathsf {T}}^1$$ T 1 an extra condition on solution