Non-degenerate hypergraphs with exponentially many extremal constructions

For every integer $t⩾0$, denote by $F_5^t$ the hypergraph on vertex set $\{1,2,\dots ,5+t\}$ with hyperedges $\{123,124\}\cup \{34k:5⩽k⩽5+t\}$. We determine $\mathrm{ex}(n,F_5^t)$ for every $t⩾0$ and sufficiently large n and characterize the extremal $F_5^t$-free hypergraphs. In particular, if n satisfies certain divisibility conditions, then the extremal $F_5^t$-free hypergraphs are exactly the balanced complete tripartite hypergraphs with additional hyperedges inside each of the three parts $(V_1,V_2,V_3)$ in the partition; each part $V_i$ spans a $\left(\right|V_i|,3,2,t)$-design. This generalizes earlier work of Frankl and Füredi on the Turán number of $F_5:=F_5^0$.

Our results extend a theory of Erdős and Simonovits about the extremal constructions for certain fixed graphs. In particular, the hypergraphs $F_5^{6t}$, for $t⩾1$, are the first examples of hypergraphs with exponentially many extremal constructions and positive Turán density.