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.