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arXiv · 2604.05288

Induced rational exponents near two

Tao Jiang·Sean Longbrake
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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.

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Tao Jiang, Sean Longbrake. 2026-04-07. Induced rational exponents near two. https://arxiv.org/abs/2604.05288

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