Rational exponents in extremal graph theory

Abstract

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

11 pages. arXiv admin note: text overlap with arXiv:1411.0856