The number of hypergraphs without linear cycles

The r-uniform linear k-cycle C_k ^r is the r-uniform hypergraph on k(r−1) vertices whose edges are sets of r consecutive vertices in a cyclic ordering of the vertex set chosen in such a way that every pair of consecutive edges share exactly one vertex. Here, we prove a balanced supersaturation result for linear cycles which we then use in conjunction with the method of hypergraph containers to show that for any fixed pair of integers r,k≥3, the number of C_k ^r-free r-uniform hypergraphs on n vertices is 2^Θ(nr−1), thereby settling a conjecture due to Mubayi and Wang from 2017.