Superlogarithmic Cliques in Dense Inhomogeneous Random Graphs

Abstract

In the theory of dense graph limits, a graphon is a symmetric measurable function W : [0, 1] 2 → [0, 1]. Each graphon gives rise naturally to a random graph distribution, denoted G(n, W ), that can be viewed as a generalization of the Erdős-Rényi random graph. Recently, Doležal, Hladký, and Máthé gave an asymptotic formula of order log n for the clique number of 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 G(n, W ) will be Θ( √ n) almost surely. We also give a family of examples with clique number Θ(n α ) for any α ∈ (0, 1), and some conditions under which the clique number of G(n, W ) will be o( √ n), ω( √ n), or Ω(n α ) for α ∈ (0, 1).

Locally uniform random permutations with large increasing subsequences

“…In some sense this is analogous to Deuschel and Zeitouni's result on permutations (Theorem 1.2 here). In [McK19] the author studies graphons allowed to approach the value 1, and proves in several cases that clique numbers behave as a power of N ; the results of the present paper are counterparts for permutations.…”

“…This paper was partly motivated by [McK19], where an analogous problem is tackled for graphons. The theory of graphons for the study of dense graph sequences is arguably the main inspiration at the origin of permuton theory, and there exist numerous bridges between them [GGKK15, BBD + 22].…”