The hard-core model in graph theory

TL;DR

This paper explores the hard-core model in graph theory, revealing how local properties of independent sets influence global graph structures and their applications in areas like Ramsey numbers and sphere packings.

Contribution

It provides new insights into the global structure of independent sets through local analysis of the hard-core model, connecting to various combinatorial problems.

Findings

Local analysis of the hard-core model yields global structural insights.

Connections established between independent sets and Ramsey numbers.

Implications for graph colorings and sphere packings.

Abstract

An independent set may not contain both a vertex and one of its neighbours. This basic fact makes the uniform distribution over independent sets rather special. We consider the hard-core model, an essential generalization of the uniform distribution over independent sets. We show how its local analysis yields remarkable insights into the global structure of independent sets in the host graph, in connection with, for instance, Ramsey numbers, graph colourings, and sphere packings.