On colorings of variable words

TL;DR

This paper establishes that the base case of the Graham--Rothschild Theorem, involving colorings of 1-dimensional variable words, has bounds within the class  of Grzegorczyk's hierarchy, indicating a certain computational complexity.

Contribution

It proves that the foundational case of the Graham--Rothschild Theorem has bounds in the  class of Grzegorczyk's hierarchy, providing new complexity insights.

Findings

Bounds are in the class  of Grzegorczyk's hierarchy.

The result applies specifically to the 1-dimensional variable words case.

This advances understanding of the theorem's computational complexity.

Abstract

In this note, we prove that the base case of the Graham--Rothschild Theorem, i.e., the one that considers colorings of the ($1$-dimensional) variable words, admits bounds in the class $\mathcal{E}^5$ of Grzegorczyk's hierarchy.