On finitary Hindman's numbers

TL;DR

This paper proves that the Paris-Harrington version of the Folkman-Sanders theorem has primitive recursive upper bounds, answering Spencer's question affirmatively.

Contribution

It establishes the primitive recursive bounds for a significant combinatorial theorem, advancing understanding of its computational complexity.

Findings

Confirmed primitive recursive bounds exist for the theorem

Provided a constructive method to obtain these bounds

Enhanced the connection between combinatorics and computational complexity

Abstract

Spencer asked whether the Paris-Harrington version of the Folkman-Sanders theorem has primitive recursive upper bounds. We give a positive answer to this question.