Extremal hypergraphs without generalized 4-cycles

Abstract

In 1977, Erdős posed the problem of determining the maximum number $f_r(n)$ of edges in an $n$-vertex $r$-uniform hypergraph in which all disjoint pairs of edges have distinct unions. Füredi later conjectured that, for every fixed $r\ge 4$ and all sufficiently large $n$, $f_r(n)=\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)+\lfloor \frac{n-1}{r}\rfloor$. In this paper, we prove this conjecture and determine all extremal configurations. Our proof combines a stability theorem for such dense hypergraphs with a delicate deletion argument applied to an associated bipartite $3$-graph. The stability theorem also resolves a conjecture of Mubayi.