Induced rational exponents near two

TL;DR

This paper proves a conjecture about the maximum edges in graphs avoiding certain induced bipartite subgraphs, specifically for rational exponents near two, extending previous results and supporting broader conjectures.

Contribution

It establishes the existence of graphs with edge counts growing as n to the rational power 2 - a/b for specific rational numbers, extending known results to the induced case.

Findings

Proved the conjecture for all rationals r=2 - a/b with specified conditions.

Extended a known result of Conlon and Janzer to the induced setting.

Provided evidence supporting the conjecture relating to rational exponents in extremal graph theory.

Abstract

Given a bipartite graph $H$ and a natural number $s$, let $\mathrm{ex}^*(n,H,s)$ denote the maximum number of edges in an $n$-vertex graph that contains neither $K_{s,s}$ nor an induced copy of $H$. Hunter, Milojevi\'c, Sudakov, and Tomon conjectured that $\mathrm{ex}^*(n,H,s)=O_{H,s}(\mathrm{ex}(n,H))$ whenever $H$ is connected. Motivated by this conjecture and the rational exponents conjecture, Dong, Gao, Li, and Liu conjectured that for every rational $r\in (1,2)$ there is a bipartite graph $H$ and an $s_0$ such that $\mathrm{ex}^*(n,H,s)=\Theta(n^r)$ for all $s\geq s_0$. We prove that the latter conjecture holds for all rationals $r=2-a/b$, where $a,b\in\mathbb{N}$ satisfy $b\geq \max\{a,(a-1)^2\}$. Our result extends a well-known result of Conlon and Janzer to the induced setting and adds more evidence to support the former conjecture.