A Ramsey Theorem for Graded Lattices

TL;DR

This paper establishes a Van der Waerden type theorem within graded lattices, demonstrating its applicability to structures like set partitions and Boolean lattices, and deriving the Hales-Jewett theorem as a consequence.

Contribution

It introduces a new axiomatic framework for graded lattices that generalizes classical combinatorial theorems.

Findings

Proves a Van der Waerden type theorem for graded lattices

Shows the framework applies to set partitions and Boolean lattices

Derives the Hales-Jewett theorem as a corollary

Abstract

We develop a Van der Waerden type theorem in an axiomatic setting of graded lattices and show that this axiomatic formulation can be applied to various lattices, for instance the set partition and the Boolean lattices. We derive the Hales-Jewett theorem as a corollary.