Turán theorems for even cycles in random hypergraph

Let $\mathbb{F}$ be an $r$-uniform hypergraph. The random Tur'an number $\mathrm{ex}(G^r_{n,p},\mathbb{F})$ is the maximum number of edges in an $\mathbb{F}$-free subgraph of $G^r_{n,p}$, where $G^r_{n,p}$ is the Erd\H{o}s-R'enyi random $r$-graph with parameter $p$. Let $C^r_{\ell}$ denote the $r$-uniform linear cycle of length $\ell$. For $p\ge n^{-r+2+o(1)}$, Mubayi and Yepremyan showed that $\mathrm{ex}(G^r_{n,p},C^r_{2\ell})\le\max{p^{\frac{1}{2\ell-1}}n^{1+\frac{r-1}{2\ell-1}+o(1)},pn^{r-1+o(1)}}$. This upper bound is not tight when $p\le n^{-r+2+\frac{1}{2\ell-2}+o(1)}$. Recently, we close the gap for $r\ge 4$. More precisely, we show that $\mathrm{ex}(G^r_{n,p},C^r_{2\ell})=\Theta(pn^{r-1})$ when $p\ge n^{-r+2+\frac{1}{2\ell-1}+o(1)}$. Similar results have recently been obtained independently in a different way by Mubayi and Yepremyan. For $r=3$, we significantly improve Mubayi and Yepremyan’s upper bound.