For an $r$-uniform hypergraph $H$ and a family of $r$-uniform hypergraphs $\mathcal{ F}$, the relative Tur\'{ a}n number $\mathrm{ ex}(H,\mathcal{ F})$ is the maximum number of edges in an $\mathcal{ F}$-free subgraph of $H$. In this paper we give lower bounds on $\mathrm{ ex}(H,\mathcal{ F})$ for certain families of hypergraph cycles $\mathcal{ F}$ such as Berge cycles and loose cycles. In particular, if $\mathcal{ C}_\ell^3$ denotes the set of all $3$-uniform Berge $\ell$-cycles and $H$ is a 3-uniform hypergraph with maximum degree $\Delta$, we prove \[\mathrm{ ex}(H,\mathcal{ C}_4^{ 3})\ge \Delta^{ -3/4-o(1)}e(H),\] \[\mathrm{ ex}(H,\mathcal{ C}_5^{ 3})\ge \Delta^{ -3/4-o(1)}e(H),\] and these bounds are tight up to the $o(1)$ term.
Categories: math.CO