Relative Turan numbers for hypergraph cycles

For an r-uniform hypergraph H and a family of r-uniform hypergraphs F, the relative Turan 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 C-l(3) denotes the set of all 3-uniform Berge l-cycles and His a 3-uniform hypergraph with maximum degree Delta, we prove ex(H, C-4(3)) >= Delta(-3/4-o(1))e(H), ex(H, C-5(3)) >= Delta(-3/4-o(1))e(H), and these bounds are tight up to the o(1) term. (C) 2021 Elsevier B.V. All rights reserved.