# co.combinatorics – making a random uniform hypergraph linear

Let $$mathcal{H}_{n,p,h}=(V,E)$$ be a random $$h$$-uniform hypergraph on $$(n)$$, sampled according to the usual binomial distribution. We known that with high probability, the number of edges in $$mathcal{H}_{n,p,h}$$ is
$$m = (1+o(1))binom{n}{h}p$$

Let $$ell$$ be given. I would like to delete some edges in order to

• have a linear hypergraph (any two edges share at most one vertex)
• remove all cycles of length at most $$ell$$

I expect that we should be able to do so by deleting with high probabilities $$o(m)$$ edges, however simple first moment method are failing me… I try to count the number of Berge-cycle of length of length at most $$ell$$, but simply looking at potential cycles for each pair of vertices I over-count way too much.

Is there any known upper bound for the number of cycles ? I found some literature on the probability threshold for the appearance of cycles, but not much on counting the cycles.