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

A 3-graph $\mathcal{F}$ is \emph{$U(s,2s+1)$} if for any $s$ edges $e_1,\mathrm{.}\mathrm{.}\mathrm{.},e_s\in E(\mathcal{F})$, $\mathrm{\mid }{e}_1\cup \mathrm{.}\mathrm{.}\mathrm{.}\cup e_s\mathrm{\mid }\le 2s+1$. 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 $$e(\mathcal{F})\le \max \{\left(\genfrac{}{}{0px}{}{n-1}{2}\right),(n-s-1)\left(\genfrac{}{}{0px}{}{s+1}{2}\right)+\left(\genfrac{}{}{0px}{}{s+1}{3}\right),\left(\genfrac{}{}{0px}{}{2s+1}{3}\right)\}\mathrm{.}$$ In this paper, we confirm Frankl and Kupavskii’s conjecture.