Relative Tur\'an Numbers for Hypergraph Cycles

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