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Tuesday, August 18, 2020

**Abstract:** For two $r$-uniform hypergraphs $G$ and $H$ the Turan number $\mathrm{ex}(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $\mathrm{ex}(G, H)$ when $G=G_{n,p}^{(r)}$, the Erdos-Renyi random $r$-uniform hypergraph for $r\geq 3$ and $H=C_{2\ell}^{(r)}$, the $r$-uniform linear cycle of length $2\ell$. The case of graphs ($r=2$) is a longstanding open problem that was studied by many researchers.

We determine $\mathrm{ex}(G_{n,p}^{(r)}, C_{2\ell}^{(r)})$ up to polylogarithmic factors for all but a small interval of values of $p=p(n)$ whose length vanishes as $\ell\rightarrow\infty$. Our main technical contribution is a balanced supersaturation result for linear even cycles which substantially improves previous results of Ferber, Samotij and Mckinley and Balogh, Narayanan and Skokan. This is joint work with Dhruv Mubayi.

Please contact Sean English at SEnglish (at) illinois (dot) edu for Zoom details.