Van der Waerden's Colouring Theorem and Weak-Strong Duality on the Lattice

TL;DR

This paper extends Van der Waerden's colouring theorem to three colours, revealing new dualities in lattice models and providing insights into complex lattice structures like ternary alloys.

Contribution

It solves the three-colour Van der Waerden problem and introduces a novel duality concept linking multicolour and monochromatic lattice mappings.

Findings

Solved the three-colour Van der Waerden problem.

Established a new duality between multicolour and monochromatic lattice mappings.

Provided insights into complex lattice structures such as ternary alloys.

Abstract

Van der Waerden's (VDW) colouring theorem in combinatoric number theory [1] has scope for physical applications.The solution of the two colour case has enabled the construction of an explicit mapping of an infinite, one dimensional antiferromagnetic Ising system to an effective pseudo- ferromagnetic one with the coupling constants in the two cases becoming related [2].Here the three colour problem is solved and the results are used to obtain new insights in the theory of complex lattices, particularly those relating to ternary alloys. The existence of these mappings of a multicolour lattice onto a monochromatic one with different couplings illustrates {\it a new form of duality}.