Relative Turán numbers for hypergraph cycles

For an r-uniform hypergraph H and a family of r-uniform hypergraphs F, the relative Turán number ex(H,F) is the maximum number of edges in an F-free subgraph of H. In this paper we give lower bounds on ex(H,F) for certain families of hypergraph cycles F such as Berge cycles and loose cycles. In particular, if Cℓ3 denotes the set of all 3-uniform Berge ℓ-cycles and H is a 3-uniform hypergraph with maximum degree Δ, we proveex(H,C43)≥Δ−3/4−o(1)e(H),ex(H,C53)≥Δ−3/4−o(1)e(H), and these bounds are tight up to the o(1) term.