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

Abstract

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.

“Weak yet strong” restrictions of Hindman’s Finite Sums Theorem

“…Recently there has been some interest in the computability-theoretic and proof-theoretic strength of restrictions of Hindman's Theorem (see [10,5,4]). While [10] deals with a restriction on the sequence of finite sets in the Finite Unions formulation of Hindman's Theorem, both [5] and [4] deal with restrictions on the types of sums that are guaranteed to be colored the same color.…”

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

“…In recent literature [3][4][5][6]10,11] restrictions of Hindman's Theorem of the following general form have been investigated. Let S be a family of finite subsets of the positive integers.…”

“…Much recent research focused on restrictions of Hindman's Theorem in which only some finite sums are required to be monochromatic; see e.g. [3][4][5][6]10,11]. Starting from [2], much attention has been paid to restrictions based on the number of terms of monochromatic sums (see [5,10,11])-we call these quantitative restrictions.…”

“…Starting from [2], much attention has been paid to restrictions based on the number of terms of monochromatic sums (see [5,10,11])-we call these quantitative restrictions. More general forms of restrictions-which we call structural restrictions-have been shown to possess interesting logical and computational properties [3][4][5][6]. For example, if one requires monochromaticity only for arbitrarily long finite sums of successive elements of an infinite set then one obtains a principle of strength roughly that of Ramsey's Theorem for pairs, see [4].…”

A Note on Hindman-Type Theorems for Uncountable Cardinals

“…We prove our Adjacent Hindman Theorem and at the same time provide an upper bound. While in the countable case the Adjacent Hindman theorem follows from Ramsey's theorem (see [1], Proposition 1), in the uncountable setting the proof hinges on the Erdős-Rado theorem. Proof.…”