Shelah's partition functions and the Hales-Jewett numbers

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.