On the number of H-free hypergraphs

Two central problems in extremal combinatorics are concerned with estimating the number ex(n,H) , the size of the largest H -free hypergraph on n vertices, and the number forb(n,H) of H -free hypergraph on n vertices. It is well known that forb(n,H)=2(1+o(1))ex(n,H) for k -uniform hypergraphs that are not k -partite. In a recent breakthrough, Ferber, McKinley, and Samotij proved that for many k -partite (or degenerate ) hypergraphs H , forb(n, H) = 2O(ex(n, H)) . However, there are few known instances of degenerate hypergraphs H for which forb(n,H)=2(1+o(1))ex(n,H) holds. In this paper, we show that forb(n,H)=2(1+o(1))ex(n,H) holds for a wide class of degenerate hypergraphs known as $2$ -contractible hypertrees. This is the first known infinite family of degenerate hypergraphs H for which forb(n,H)=2(1+o(1))ex(n,H) holds. As a corollary of our main results, we obtain a sharp estimate of forb(n,C⁽ᵏ⁾_ )=2^( -1/2 +o(1)) nk-1 for the k -uniform linear -cycle, for all pairs k≥ 5, ≥ 3 , thus settling a question of Balogh, Narayanan, and Skokan affirmatively for all k≥ 5, ≥ 3 . Our methods also lead to some sharp results on the related random Turán problem. As a key ingredient of our proofs, we develop a novel supersaturation variant of the delta systems method for set systems, which may be of independent interest.