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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.19621 |
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| _version_ | 1866918168980619264 |
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| author | van der Beek, Bas Bishnoi, Anurag |
| author_facet | van der Beek, Bas Bishnoi, Anurag |
| contents | The generalized Turán number $\text{ex}(n,H,\mathcal{F})$ denotes the maximum number of copies of $H$ in an $n$-vertex graph which contains no copies of any graph in a family $\mathcal{F}$ of graphs. The generalized rational exponents conjecture states that for every rational $r\geq 1$ there exist graphs $H,F$ such that $\text{ex}(n,H,\{F\})=Θ(n^r)$. We extend a result of Bukh and Conlon to show that for every non-empty graph $H$ on $v\geq 2$ vertices and every rational $r$ in the interval $[v-1,v]$ there exists a finite family $\mathcal{F}_r$ such that $\text{ex}(n,H,\mathcal{F}_r)=Θ(n^r)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19621 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rational Exponents for Generalized Turán Numbers van der Beek, Bas Bishnoi, Anurag Combinatorics 05D40, 05E14 The generalized Turán number $\text{ex}(n,H,\mathcal{F})$ denotes the maximum number of copies of $H$ in an $n$-vertex graph which contains no copies of any graph in a family $\mathcal{F}$ of graphs. The generalized rational exponents conjecture states that for every rational $r\geq 1$ there exist graphs $H,F$ such that $\text{ex}(n,H,\{F\})=Θ(n^r)$. We extend a result of Bukh and Conlon to show that for every non-empty graph $H$ on $v\geq 2$ vertices and every rational $r$ in the interval $[v-1,v]$ there exists a finite family $\mathcal{F}_r$ such that $\text{ex}(n,H,\mathcal{F}_r)=Θ(n^r)$. |
| title | Rational Exponents for Generalized Turán Numbers |
| topic | Combinatorics 05D40, 05E14 |
| url | https://arxiv.org/abs/2510.19621 |