Induced rational exponents near two
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918482052907008 |
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| author | Jiang, Tao Longbrake, Sean |
| author_facet | Jiang, Tao Longbrake, Sean |
| contents | 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, Milojević, 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 $\mathrm{ex}^*(n,H,s)=Θ(n^r)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05288 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Induced rational exponents near two Jiang, Tao Longbrake, Sean Combinatorics 05C35 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, Milojević, 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 $\mathrm{ex}^*(n,H,s)=Θ(n^r)$ 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. |
| title | Induced rational exponents near two |
| topic | Combinatorics 05C35 |
| url | https://arxiv.org/abs/2604.05288 |