Van der Waerden's Colouring Theorem and Classical Spin Systems

TL;DR

This paper introduces a matrix representation of Van der Waerden's coloring theorem for two colors, and maps a 1D antiferromagnetic Ising system to a pseudo-ferromagnetic system, revealing a renormalization-like relationship.

Contribution

It provides a novel matrix representation for Van der Waerden's theorem and establishes a mapping between antiferromagnetic and pseudo-ferromagnetic Ising systems.

Findings

Matrix representation of Van der Waerden's theorem for two colors

Mapping of 1D antiferromagnetic to pseudo-ferromagnetic Ising system

Relation between couplings reminiscent of renormalization group

Abstract

We find a non-invertible matrix representation for Van der Waerden's colouring theorem for two distinct colours in a one dimensional periodic lattice. Using this,an infinite one dimensional antiferromagnetic Ising system is mapped to a pseudo-ferromagnetic one, thereby relating the couplings. All this is reminiscent of renormalisation group.