Finite Hypergraph Families with Rich Extremal Turán Constructions via Mixing Patterns

Abstract

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.

revised according to referee's suggestions

References in corpus (3)

• Hypergraphs with many extremal configurations
• Hypergraphs with infinitely many extremal constructions
• Non-degenerate Hypergraphs with Exponentially Many Extremal Constructions