Rational exponents for cliques

TL;DR

This paper explores the possible asymptotic growth rates of the maximum number of copies of a fixed graph in large graphs avoiding certain subgraphs, showing that all rational exponents between 1 and t are achievable for complete graphs K_t.

Contribution

It generalizes previous results to show all rational exponents between 1 and t are realizable for K_t, extending the understanding of extremal graph configurations.

Findings

Every rational between 1 and t is realizable for K_t.

Determined the realizable rationals for star graphs.

Connected the problem to Sidorenko-type supersaturation issues.

Abstract

Let $\mathrm{ex}(n,H,\mathcal{F})$ be the maximum number of copies of $H$ in an $n$-vertex graph which contains no copy of a graph from $\mathcal{F}$. Thinking of $H$ and $\mathcal{F}$ as fixed, we study the asymptotics of $\mathrm{ex}(n,H,\mathcal{F})$ in $n$. We say that a rational number $r$ is \emph{realizable for $H$} if there exists a finite family $\mathcal{F}$ such that $\mathrm{ex}(n,H,\mathcal{F}) = \Theta(n^r)$. Using randomized algebraic constructions, Bukh and Conlon showed that every rational between $1$ and $2$ is realizable for $K_2$. We generalize their result to show that every rational between $1$ and $t$ is realizable for $K_t$, for all $t \geq 2$. We also determine the realizable rationals for stars and note the connection to a related Sidorenko-type supersaturation problem.