Shelah's partition functions and the Hales-Jewett numbers

TL;DR

This paper explores Shelah's partition relations and their connection to Hales-Jewett numbers, providing an improved upper bound using a primitive recursive function within the Grzegorczyk hierarchy.

Contribution

It establishes a new upper bound for Hales-Jewett numbers using a specific primitive recursive function, refining previous results.

Findings

Upper bound for Hales-Jewett numbers using $^{8,*}$

The function $^{8,*}$ belongs to class $^5$ of the Grzegorczyk hierarchy

The new bound grows slower than $^{13}$, improving prior bounds.

Abstract

In this paper we study several partition relations, defined by Saharon Shelah, and relate them to the Hales-Jewett numbers. In particular we give an upper bound for the Hales-Jewett numbers using the primitive recursive function $\mathtt{f}^{8,*}$ which belongs to the class $\mathcal{E}^5$ of the Grzegorczyk hierarchy and grows slower than the function $\mathtt{f}^{13}$. This improves the recent result of the first author and Shelah.