On the connectivity of random subsets of projective spaces Original Research Article

It is shown that with probability tending to 1 as r → ∞ a random subset ωr(n) of n elements of the projective space PG(r − 1, q) becomes k-connected when n = r + (k − 1) logq r + O(1). Furthermore, the cyclic structure of ωr(n) is considered, and a similar very sharp threshold for the existence of circuits of length r is found.