Counting hypergraphs with large girth

Morris and Saxton used the method of containers to bound number n-vertex graphs with m edges containing no ℓ-cycles, hence girth more than ℓ. We consider a generalization r-uniform hypergraphs. The hypergraph H is minimum ℓ ≥ 2 such that there exist distinct vertices v 1 , … hyperedges e i + ∈ for all ≤ Letting N r ( n ) denote hypergraphs larger defining λ = ⌈ − ∕ ⌉ we show which tight when divides up o term in exponent. This result address extremal problem subgraphs random