Non-degenerate Hypergraphs with Exponentially Many Extremal Constructions

TL;DR

This paper determines the maximum size of hypergraphs avoiding certain structures, characterizes all extremal configurations, and introduces hypergraphs with exponentially many extremal constructions and positive Turán density.

Contribution

It generalizes previous results by characterizing extremal hypergraphs for a family of structures and introduces the first hypergraphs with exponentially many extremal configurations.

Findings

Exact Turán numbers for all t and large n.

Characterization of extremal F_5^t-free hypergraphs.

Existence of hypergraphs with exponentially many extremal constructions.

Abstract

For every integer $t \ge 0$, denote by $F_5^t$ the hypergraph on vertex set $\{1,2,\ldots, 5+t\}$ with hyperedges $\{123,124\} \cup \{34k : 5 \le k \le 5+t\}$. We determine $\mathrm{ex}(n,F_5^t)$ for every $t\ge 0$ 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 $(|V_i|,3,2,t)$-design. This generalizes earlier work of Frankl and F\"uredi on the Tur\'an number of $F_5:=F_5^0$. Our results extend a theory of Erd\H{o}s and Simonovits about the extremal constructions for certain fixed graphs. In particular, the hypergraphs $F_5^{6t}$, for $t\geq 1$, are the first examples of hypergraphs with exponentially many extremal constructions and positive Tur\'an density.