  Let Delta_{n-1} denote the (n-1)-dimensional simplex. Let Y be a random
k-dimensional subcomplex of Delta_{n-1} obtained by starting with the full
(k-1)-dimensional skeleton of Delta_{n-1} and then adding each k-simplex
independently with probability p. Let H_{k-1}(Y;R) denote the (k-1)-dimensional
reduced homology group of Y with coefficients in a finite abelian group R. Let
R and k \geq 1 be fixed. It is shown that p=(k \log n)/n is a sharp threshold
for the vanishing of H_{k-1}(Y;R).
