Cliques in dense inhomogeneous random graphs

TL;DR

This paper investigates the clique number in inhomogeneous dense random graphs generated by graphons, providing concentration results, asymptotic formulas, and insights into graphons avoiding fixed subgraphs.

Contribution

It establishes the almost sure asymptotic clique number for inhomogeneous dense graphs and explores their long-term behavior, including bipartite cases and structural properties of graphons.

Findings

Concentration of clique number for fixed n

Asymptotic formula for clique number

Graphons avoiding fixed graphs are countably-partite

Abstract

The theory of dense graph limits comes with a natural sampling process which yields an inhomogeneous variant G(n,W) of the Erdos-Renyi random graph. Here we study the clique number of these random graphs. We establish the concentration of the clique number of G(n,W) for each fixed n, and give examples of graphons for which G(n,W) exhibits wild long-term behavior. Our main result is an asymptotic formula which gives the almost sure clique number of these random graphs. We obtain a similar result for the bipartite version of the problem. We also make an observation that might be of independent interest: Every graphon avoiding a fixed graph is countably-partite.