Random Turán Problems for Hypergraph Expansions
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909418906451968 |
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| author | Nie, Jiaxi Spiro, Sam |
| author_facet | Nie, Jiaxi Spiro, Sam |
| contents | Given an $r_0$-uniform hypergraph $F$, we define its $r$-uniform expansion $F^{(r)}$ to be the hypergraph obtained from $F$ by inserting $r-r_0$ distinct vertices into each edge of $F$, and we define $\mathrm{ex}(G_{n,p}^r,F^{(r)})$ to be the largest $F^{(r)}$-free subgraph of the random hypergraph $G_{n,p}^r$.
We initiate the first systematic study of $\mathrm{ex}(G_{n,p}^r,F^{(r)})$ for general hypergraphs $F$. Our main result essentially resolves this problem for large $r$ by showing that $\mathrm{ex}(G_{n,p}^r,F^{(r)})$ goes through three predictable phases whenever $F$ is Sidorenko and $r$ is sufficiently large, with the behavior of $\mathrm{ex}(G_{n,p}^r,F^{(r)})$ being provably more complex whenever $F$ has no Sidorenko expansion. Moreover, our methods unify and generalize almost all previously known results for the random Turán problem for degenerate hypergraphs of uniformity at least 3. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03406 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Random Turán Problems for Hypergraph Expansions Nie, Jiaxi Spiro, Sam Combinatorics 05D40, 05C35, 05C65, 05C80 Given an $r_0$-uniform hypergraph $F$, we define its $r$-uniform expansion $F^{(r)}$ to be the hypergraph obtained from $F$ by inserting $r-r_0$ distinct vertices into each edge of $F$, and we define $\mathrm{ex}(G_{n,p}^r,F^{(r)})$ to be the largest $F^{(r)}$-free subgraph of the random hypergraph $G_{n,p}^r$. We initiate the first systematic study of $\mathrm{ex}(G_{n,p}^r,F^{(r)})$ for general hypergraphs $F$. Our main result essentially resolves this problem for large $r$ by showing that $\mathrm{ex}(G_{n,p}^r,F^{(r)})$ goes through three predictable phases whenever $F$ is Sidorenko and $r$ is sufficiently large, with the behavior of $\mathrm{ex}(G_{n,p}^r,F^{(r)})$ being provably more complex whenever $F$ has no Sidorenko expansion. Moreover, our methods unify and generalize almost all previously known results for the random Turán problem for degenerate hypergraphs of uniformity at least 3. |
| title | Random Turán Problems for Hypergraph Expansions |
| topic | Combinatorics 05D40, 05C35, 05C65, 05C80 |
| url | https://arxiv.org/abs/2408.03406 |