Sidorenko Hypergraphs and Random Turán Numbers
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909653176156160 |
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| author | Nie, Jiaxi Spiro, Sam |
| author_facet | Nie, Jiaxi Spiro, Sam |
| contents | Let $\mathrm{ex}(G_{n,p}^r,F)$ denote the maximum number of edges in an $F$-free subgraph of the random $r$-uniform hypergraph $G_{n,p}^r$, and let $s(F):=\sup\{s: \exists H,\ t_F(H)=t_{K_r^r}(H)^{s+e(F)}>0\}$. Following recent work of Conlon, Lee, and Sidorenko, we prove non-trivial lower bounds on $\mathrm{ex}(G_{n,p}^r,F)$ whenever $s(F)>0$, i.e. $F$ is not Sidorenko. This connection between Sidorenko's conjecture and random Turán problems gives new lower bounds on $\mathrm{ex}(G_{n,p}^r,F)$ whenever $s(F)>0$, and further allows us to establish upper bounds for $s(F)$ whenever upper bounds for $\mathrm{ex}(G_{n,p}^r,F)$ are known. As a consequence, we prove that $s(\mathrm{E}^r(K_{k+1}^k))=\frac{1}{r-k}$ where $\mathrm{E}^r(K_{k+1}^k)$ is the $r$-expansion of $K_{k+1}^k$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_12873 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sidorenko Hypergraphs and Random Turán Numbers Nie, Jiaxi Spiro, Sam Combinatorics 05D40, 05C35, 05C65, 05C80 Let $\mathrm{ex}(G_{n,p}^r,F)$ denote the maximum number of edges in an $F$-free subgraph of the random $r$-uniform hypergraph $G_{n,p}^r$, and let $s(F):=\sup\{s: \exists H,\ t_F(H)=t_{K_r^r}(H)^{s+e(F)}>0\}$. Following recent work of Conlon, Lee, and Sidorenko, we prove non-trivial lower bounds on $\mathrm{ex}(G_{n,p}^r,F)$ whenever $s(F)>0$, i.e. $F$ is not Sidorenko. This connection between Sidorenko's conjecture and random Turán problems gives new lower bounds on $\mathrm{ex}(G_{n,p}^r,F)$ whenever $s(F)>0$, and further allows us to establish upper bounds for $s(F)$ whenever upper bounds for $\mathrm{ex}(G_{n,p}^r,F)$ are known. As a consequence, we prove that $s(\mathrm{E}^r(K_{k+1}^k))=\frac{1}{r-k}$ where $\mathrm{E}^r(K_{k+1}^k)$ is the $r$-expansion of $K_{k+1}^k$. |
| title | Sidorenko Hypergraphs and Random Turán Numbers |
| topic | Combinatorics 05D40, 05C35, 05C65, 05C80 |
| url | https://arxiv.org/abs/2309.12873 |