Relative Turán numbers for hypergraph cycles

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