Extremal 3-uniform hypergraphs without generalized 4-cycles

Abstract

Let $f_r(n)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph in which all disjoint pairs of edges have distinct unions. In 1977, Erdős posed the problem of determining $f_r(n)$ and, in particular, asked whether $f_3(n)=O(n^2)$. Füredi later proved that $\binom{n-1}{r-1}+\left\lfloor (n-1)/r\right\rfloor \le f_r(n)<\frac72\binom{n}{r-1}$ for $r\ge3$ and $n\ge2r$. Motivated by his constructions, he further conjectured that $f_3(n)\le\binom{n}{2}$ for all sufficiently large $n$, and that, for each fixed $r\ge4$, the lower bound is exact for all sufficiently large $n$. This is known as the Erdős--Füredi conjecture. In a recent breakthrough, Huang, Ma and Yang proved the conjecture for every fixed $r\ge4$ and characterized all extremal hypergraphs. In this paper, we settle the remaining $3$-uniform case for every $n$. The bound is sharp whenever $n\ge5$ and $n\equiv1,5\pmod{20}$. Our proof uses a local weight inequality and a transfer argument.