# Rational exponents in extremal graph theory

### Boris Bukh

Carnegie Mellon University, Pittsburgh, USA### David Conlon

University of Oxford, UK

## Abstract

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

## Cite this article

Boris Bukh, David Conlon, Rational exponents in extremal graph theory. J. Eur. Math. Soc. 20 (2018), no. 7, pp. 1747–1757

DOI 10.4171/JEMS/798