On Tur\'an exponents of bipartite graphs

TL;DR

This paper advances the understanding of Turán exponents for bipartite graphs by establishing new rational exponents, including a novel class, and explores related conjectures and bounds in extremal graph theory.

Contribution

It proves new Turán exponents for bipartite graphs, including a previously unverified class, and provides bounds for theta-graphs, contributing to the broader Erdős–Simonovits conjecture.

Findings

Established Turán exponents for the form 2-2/(2k+1) for k≥2

Derived an upper bound for Turán's number of theta-graphs in asymmetric cases

Identified a new rational exponent r=7/5 for Turán problems

Abstract

A long-standing conjecture of Erd\H{o}s and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph $H$ such that $\ex(n,H)=\Theta(n^r)$. So far this conjecture is known to be true only for rationals of form $1+1/k$ and $2-1/k$, for integers $k\geq 2$. In this paper we add a new form of rationals for which the conjecture is true; $2-2/(2k+1)$, for $k\geq 2$. This in its turn also gives an affirmative answer to a question of Pinchasi and Sharir on cube-like graphs. Recently, a version of Erd\H{o}s and Simonovits's conjecture where one replaces a single graph by a family, was confirmed by Bukh and Conlon. They proposed a construction of bipartite graphs which should satisfy Erd\H{o}s and Simonovits's conjecture. Our result can also be viewed as a first step towards verifying Bukh and Conlon's conjecture. We also prove the an upper bound on the Tur\'an's number of $\theta$-graphs in an asymmetric setting and employ this result to obtain yet another new rational exponent for Tur\'an exponents; $r=7/5$.