Rational exponents near two

TL;DR

This paper advances the understanding of extremal graph theory by confirming a conjecture for a broad class of rational exponents near two, specifically those of the form 2 - a/b with large denominators.

Contribution

It proves the Erd ext{"o}s-Simonovits conjecture for all rationals of the form 2 - a/b with sufficiently large b, answering a question posed by Jiang, Jiang, and Ma.

Findings

The conjecture holds for all rationals of the form 2 - a/b with large b.

The result extends the class of rational exponents for which the extremal function is known.

Provides new techniques for analyzing extremal functions near the exponent 2.

Abstract

A longstanding conjecture of Erd\H{o}s 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 $\Theta(n^r)$. 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 $a$.