On the number of linear hypergraphs of large girth

An r-uniform linear cycle of length l, denoted by C^r _l, is an r-graph with edges e₁…e_l such that for every i ∈ [ℓ − 1], ∣e_i ∩ e_i+1| = 1, |e_ℓ ∩ e₁| = 1, and e_i ∩ e_j = ∅ for all other pairs {i, j}, i ≠ j. For every r ≥ 3 and ℓ ≥ 4, we show that there exists a constant C depending on r and l such that the number of linear r-graphs of girth l is at most (Formula presented.). Furthermore, we extend the result for l=4, proving that there exists a constant C depending on r such that the number of linear r -graphs without C^r ₄ is at most (Formula presented.). The idea of the proof is to reduce the hypergraph enumeration problems to some graph enumeration problems, and then apply a variant of the graph container method, which may be of independent interest. We extend a breakthrough result of Kleitman and Winston on the number of C₄ -free graphs, proving that the number of graphs containing at most n² 32 log⁶ n C₄'s is at most (Formula presented.), for sufficiently large n. We further show that for every r ≥ 3 and ℓ ≥ 2, the number of graphs such that each of its edges is contained in only O(1) cycles of length at most 2l, is bounded by (Formula presented.) asymptotically.