# Journal of the European Mathematical Society

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**Volume 20, Issue 7, 2018, pp. 1747–1757**

**DOI: 10.4171/JEMS/798**

Published online: 2018-05-22

Rational exponents in extremal graph theory

Boris Bukh^{[1]}and David Conlon

^{[2]}(1) Carnegie Mellon University, Pittsburgh, USA

(2) University of Oxford, UK

Given a family of graphs $\mathcal{H}$, the extremal number 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 ex$(n, \mathcal{H}_r) = \Theta(n^r)$. This solves a longstanding problem in the area of extremal graph theory.

*Keywords: *Extremal graph theory, bipartite graphs, algebraic constructions

Bukh Boris, Conlon David: Rational exponents in extremal graph theory. *J. Eur. Math. Soc.* 20 (2018), 1747-1757. doi: 10.4171/JEMS/798