A finite partition theorem with double exponential bounds

TL;DR

This paper establishes that double exponential bounds are sufficient for a specific Ramsey theorem involving pair colourings and the order of differences in homogeneous sets.

Contribution

It introduces a finite partition theorem with explicit double exponential bounds for a particular Ramsey-type problem.

Findings

Double exponential bounds are proven to be an upper limit.

The theorem applies to colouring pairs with constraints on differences.

Provides a quantitative bound for a Ramsey-type combinatorial problem.

Abstract

We prove that double exponentiation is an upper bound to Ramsey theorem for colouring of pairs when we want to predetermine the order of the differences of successive members of the homogeneous set.