The binomial random graph is a bad inducer

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.