Rational exponents in extremal graph theory

TL;DR

This paper demonstrates that for every rational exponent between 1 and 2, there exists a graph family with extremal number growing as a power of n with that exponent, resolving a major open problem.

Contribution

The authors construct specific graph families for each rational exponent between 1 and 2, showing the extremal number can have any such growth rate, solving a longstanding problem.

Findings

For every rational r in (1,2), a graph family with extremal number Θ(n^r) exists.

The result bridges a gap in understanding extremal functions for rational exponents.

This work confirms the realizability of all rational growth rates in extremal graph theory.

Abstract

Given a family of graphs $\mathcal{H}$, the extremal number $\textrm{ex}(n, \mathcal{H})$ is the largest $m$ for which there exists a graph with $n$ vertices and $m$ edges containing no graph from the family $\mathcal{H}$ as a subgraph. We show that for every rational number $r$ between $1$ and $2$, there is a family of graphs $\mathcal{H}_r$ such that $\textrm{ex}(n, \mathcal{H}_r) = \Theta(n^r)$. This solves a longstanding problem in the area of extremal graph theory.