Rational exponents near two

Abstract

A longstanding conjecture of Erdős and Simonovits states that for every rational $r$ between $1$ and $2$ there is a graph $H$ such that the largest number of edges in an $H$-free graph on $n$ vertices is $\hat{I}˜(n^r)$. Answering a question raised by Jiang, Jiang and Ma, we show that the conjecture holds for all rationals of the form $2-a/b$ with $b$ sufficiently large in terms of $a$.

10 pages