A proof of Frankl–Kupavskii's conjecture on edge‐union condition.

A 3‐graph F ${\rm{ {\mathcal F} }}$ is U(s,2s+1) $U(s,2s+1)$ if for any s $s$ edges e1,...,es∈E(F) ${e}_{1},\ldots ,{e}_{s}\in E({\rm{ {\mathcal F} }})$, ∣e1∪⋯∪es∣≤2s+1 $| {e}_{1}\cup \cdots \cup {e}_{s}| \le 2s+1$. Frankl and Kupavskii proposed the following conjecture: For any 3‐graph F ${\rm{ {\mathcal F} }}$ with n $n$ vertices, if F ${\rm{ {\mathcal F} }}$ is U(s,2s+1) $U(s,2s+1)$, then e(F)≤maxn−12,(n−s−1)s+12+s+13,2s+13. $\begin{array}{c}e({\mathscr{F}})\le \max \left\{\left(\genfrac{}{}{0ex}{}{n-1}{2}\right),(n-s-1)\left(\genfrac{}{}{0ex}{}{s+1}{2}\right)\right.\\ \,\left.+\left(\genfrac{}{}{0ex}{}{s+1}{3}\right),\left(\genfrac{}{}{0ex}{}{2s+1}{3}\right)\right\}.\end{array}$ In this paper, we confirm Frankl and Kupavskii's conjecture. [ABSTRACT FROM AUTHOR]