Finite Hypergraph Families with Rich Extremal Turán Constructions via Mixing Patterns
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916648631402496 |
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| author | Liu, Xizhi Pikhurko, Oleg |
| author_facet | Liu, Xizhi Pikhurko, Oleg |
| contents | We prove that, for any finite set of minimal $r$-graph patterns, there is a finite family $\mathcal F$ of forbidden $r$-graphs such that the extremal Turán constructions for $\mathcal F$ are precisely the maximum $r$-graphs obtainable from mixing the given patterns in any way via blowups and recursion. This extends the result by the second author \cite{PI14}, where the above statement was established for a single pattern.
We present two applications of this result. First, we construct a finite family $\mathcal F$ of $3$-graphs such that there are exponentially many maximum $\mathcal F$-free $3$-graphs of each large order $n$ and, moreover, the corresponding Turán problem is not finitely stable. Second, we show that there exists a finite family $\mathcal{F}$ of $3$-graphs whose feasible region function attains its maximum on a Cantor-type set of positive Hausdorff dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_08636 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Finite Hypergraph Families with Rich Extremal Turán Constructions via Mixing Patterns Liu, Xizhi Pikhurko, Oleg Combinatorics We prove that, for any finite set of minimal $r$-graph patterns, there is a finite family $\mathcal F$ of forbidden $r$-graphs such that the extremal Turán constructions for $\mathcal F$ are precisely the maximum $r$-graphs obtainable from mixing the given patterns in any way via blowups and recursion. This extends the result by the second author \cite{PI14}, where the above statement was established for a single pattern. We present two applications of this result. First, we construct a finite family $\mathcal F$ of $3$-graphs such that there are exponentially many maximum $\mathcal F$-free $3$-graphs of each large order $n$ and, moreover, the corresponding Turán problem is not finitely stable. Second, we show that there exists a finite family $\mathcal{F}$ of $3$-graphs whose feasible region function attains its maximum on a Cantor-type set of positive Hausdorff dimension. |
| title | Finite Hypergraph Families with Rich Extremal Turán Constructions via Mixing Patterns |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2212.08636 |