Homological connectivity of random 2-complexes

Let Δ _n-1 denote the (n - 1)-dimensional simplex. Let Y be a random 2-dimensional subcomplex of Δ _n-1 obtained by starting with the full 1-dimensional skeleton of Δ _n-1 and then adding each 2-simplex independently with probability p. Let H ₁(Y;₂) denote the first homology group of Y with mod 2 coefficients. It is shown that for any function ω(n) that tends to infinity lim_{n →∞} Prob[H₁(Y;double-struck F sign₂) = 0{0 p = 2 log n-w(n)/n 1 p = 2 log n+w(n)/n.