A recursive coloring function without $\Pi_3^0$ solution for Hindman theorem

TL;DR

This paper constructs recursive coloring functions demonstrating the limitations of $ ext{Pi}_3^0$ and $ ext{Delta}_3^0$ sets as solutions to Hindman's theorem, highlighting the theorem's complexity.

Contribution

It introduces recursive coloring functions that cannot be solved by any $ ext{Pi}_3^0$ or $ ext{Delta}_3^0$ sets, revealing new complexity boundaries in Hindman's theorem.

Findings

Existence of recursive colorings not solvable by $ ext{Pi}_3^0$ sets.

Existence of recursive colorings not solvable by $ ext{Delta}_3^0$ sets.

Limits on the solutions to Hindman's theorem in the arithmetical hierarchy.

Abstract

We show that there exists a recursive coloring function $c$ such that any $\Pi^0_3$ set is not a solution to $c$ for Hindman's theorem. We also show that there exists a recursive coloring function $c$ such that any $\Delta^0_3$ set is not a solution to $c$ for Hindman's theorem restricted on sum of at most three numbers.