A proof of Frankl-Kupavskii's conjecture on edge-union condition

TL;DR

This paper proves Frankl and Kupavskii's conjecture, establishing an upper bound on the number of edges in 3-graphs satisfying a specific union condition, thus resolving a significant open problem in hypergraph theory.

Contribution

The paper provides a rigorous proof confirming the conjecture that bounds edges in 3-graphs under the $U(s, 2s+1)$ condition, advancing understanding in extremal hypergraph problems.

Findings

Confirmed Frankl and Kupavskii's conjecture.

Established exact upper bounds for edges in $U(s, 2s+1)$ 3-graphs.

Resolved an open problem in hypergraph extremal theory.

Abstract

A 3-graph $\mathcal{F}$ is \emph{$U(s, 2s+1)$} if for any $s$ edges $e_1,...,e_s\in E(\mathcal{F})$, $|e_1\cup...\cup e_s|\leq 2s+1$. Frankl and Kupavskii (2020) proposed the following conjecture: For any $3$-graph $\mathcal{F}$ with $n$ vertices, if $\mathcal{F}$ is $U(s, 2s+1)$, then $$e(\mathcal{F})\leq \max\left\{{n-1\choose 2}, (n-s-1){s+1\choose 2}+{s+1\choose 3}, {2s+1\choose 3}\right\}.$$ In this paper, we confirm Frankl and Kupavskii's conjecture.