Rational Exponents for General Graphs

TL;DR

This paper characterizes which rational exponents are realizable for counting subgraphs in graphs, extending known results from stars and cliques to arbitrary graphs with bounded degree.

Contribution

It establishes the first broad set of realizable exponents for general graphs, showing they form specific intervals depending on graph parameters, and introduces a new Helly theorem variant for trees.

Findings

Rational exponents in certain intervals are realizable for graphs with bounded degree.

Trees with more than two vertices have no realizable exponents in certain non-integer intervals.

New Helly theorem variant for trees may have independent interest.

Abstract

A rational number $r$ is a \textbf{realizable exponent} for a graph $H$ if there exists a finite family of graphs $\mathcal{F}$ such that $\mathrm{ex}(n,H,\mathcal{F})=\Theta(n^r)$, where $\mathrm{ex}(n,H,\mathcal{F})$ denotes the maximum number of copies of $H$ that an $n$-vertex $\mathcal{F}$-free graph can have. Results for realizable exponents are currently known only when $H$ is either a star or a clique, with the full resolution of the $H=K_2$ case being a major breakthrough of Bukh and Conlon. In this paper, we establish the first set of results for realizable exponents which hold for arbitrary graphs $H$ by showing that for any graph $H$ with maximum degree $\Delta \ge 1$, every rational in the interval $\left[v(H)-\frac{e(H)}{2\Delta^2},\ v(H)\right]$ is realizable for $H$. We also prove a ``stability'' result for generalized Tur\'an numbers of trees which implies that if $T\ne K_2$ is a tree with $\ell$ leaves, then $T$ has no realizable exponents in $[0,\ell]\setminus \mathbb{Z}$. Our proof of this latter result uses a new variant of the classical Helly theorem for trees, which may be of independent interest.