Consider an $N$-element set $X$ and a fixed number $k \ll N$.
How many $k$-element subsets $X_i$ (hyperedges) of $X$ do I have to choose (at random) such that (i) $\bigcup_i X_i = X$ and (ii) the intersection graph (with an edge between $X_i$ and $X_j$ when $X_i \cap X_j \neq \emptyset$) is almost surely connected?
I assume this question reduces to the question for the probability that two random $k$-element subsets of $X$ have a common element. Assume that this probability is $p = p(N,k)$. Then if $p > \frac {(1+\varepsilon )\ln n}{n}$ the intersection graph is almost surely connected, where $n$ is the number I am looking for. But what is $p$?
A more advanced version of this question is: Given a Poisson distribution of hyperedge sizes with mean $k$, how many do I have to choose such that (i) and (ii) above hold?
A follow-up question is this: How does the expected size of the intersection of two hyperedges grow with the number $n$ of hyperedges? (Consider the intersection graph as a weighted graph with edge weights = size of intersection.)