Lower bounds for Ramsey numbers as a statistical physics problem

TL;DR

This paper connects Ramsey number lower bounds to a statistical physics problem, using Monte Carlo methods to estimate bounds and discussing potential for scaling to larger graphs.

Contribution

It introduces a novel Hamiltonian formulation linking Ramsey theory to statistical physics and demonstrates a Monte Carlo approach to estimate lower bounds.

Findings

Monte Carlo methods produce bounds consistent with known values

Hamiltonian design encodes Ramsey properties

Discussion of numerical limitations and future scaling

Abstract

Ramsey's theorem, concerning the guarantee of certain monochromatic patterns in large enough edge-coloured complete graphs, is a fundamental result in combinatorial mathematics. In this work, we highlight the connection between this abstract setting and a statistical physics problem. Specifically, we design a classical Hamiltonian that favours configurations in a way to establish lower bounds on Ramsey numbers. As a proof of principle we then use Monte Carlo methods to obtain such lower bounds, finding rough agreement with known literature values in a few cases we investigated. We discuss numerical limitations of our approach and indicate a path towards the treatment of larger graph sizes.