The binomial random graph is a bad inducer

TL;DR

This paper demonstrates that the binomial random graph does not maximize induced subgraph densities for graphs with at least three vertices, using graphon analysis to show it is not a local maximum in the space of graphons.

Contribution

It proves that the binomial random graph is never a local maximum for induced subgraph densities in the graphon space, answering a previously open problem.

Findings

Binomial random graph has induced F-density less than the maximum for all F with ≥3 vertices.

The binomial random graph is not a local maximum in the graphon space.

The approach uses perturbations in the graphon setting to establish the result.

Abstract

For a finite graph $F$ and a value $p \in [0,1]$, let $I(F,p)$ denote the largest $y$ for which there is a sequence of graphs of edge density approaching $p$ so that the induced $F$-density of the sequence approaches $y$. We show that for all $F$ on at least three vertices and all $p \in (0,1)$, the binomial random graph $G(n,p)$ has induced $F$-density strictly less than $I(F,p).$ This provides a negative answer to a problem posed by Liu, Mubayi and Reiher. Our approach is in the limiting setting of graphons, and we in fact show a stronger result: the binomial random graph is never a \emph{local} maximum in the space of graphons of edge density $p$. This is done by finding a sequence of balanced perturbations of arbitrarily small norm that increase the $F$-density.