  Consider a uniform expanders family G_n with a uniform bound on the degrees.
It is shown that for any p and c>0, a random subgraph of G_n obtained by
retaining each edge, randomly and independently, with probability p, will have
at most one cluster of size at least c|G_n|, with probability going to one,
uniformly in p. The method from Ajtai, Komlos and Szemeredi [Combinatorica 2
(1982) 1-7] is applied to obtain some new results about the critical
probability for the emergence of a giant component in random subgraphs of
finite regular expanding graphs of high girth, as well as a simple proof of a
result of Kesten about the critical probability for bond percolation in high
dimensions. Several problems and conjectures regarding percolation on finite
transitive graphs are presented.
