# Super-logarithmic cliques in dense inhomogeneous random graphs

## TL;DR

This paper investigates the asymptotic behavior of the clique number in inhomogeneous random graphs generated by graphons, revealing super-logarithmic growth under certain conditions and providing examples with various polynomial growth rates.

## Contribution

It extends previous results by characterizing the clique number for graphons approaching 1 at finitely many points, showing it can grow as fast as (
) and providing a broad spectrum of growth behaviors.

## Key findings

- Clique number can be (
) when W approaches 1 at finitely many points.
- Examples of graphons with clique number (n^) for any  in (0,1).
- Conditions for clique number to be o((
)), (
), or (n^).

## Abstract

In the theory of dense graph limits, a graphon is a symmetric measurable function $W:[0,1]^2\to [0,1]$. Each graphon gives rise naturally to a random graph distribution, denoted $\mathbb{G}(n,W)$, that can be viewed as a generalization of the Erd\H{o}s-R\'enyi random graph. Recently, Dole\v{z}al, Hladk\'y, and M\'ath\'e gave an asymptotic formula of order $\log n$ for the clique number of $\mathbb{G}(n,W)$ when $W$ is bounded away from 0 and 1. We show that if $W$ is allowed to approach 1 at a finite number of points, and displays a moderate rate of growth near these points, then the clique number of $\mathbb{G}(n,W)$ will be $\Theta(\sqrt{n})$ almost surely. We also give a family of examples with clique number $\Theta(n^\alpha)$ for any $\alpha\in(0,1)$, and some conditions under which the clique number of $\mathbb{G}(n,W)$ will be $o(\sqrt{n})$, $\omega(\sqrt{n}),$ or $\Omega(n^\alpha)$ for $\alpha\in(0,1)$.