  The classical result of Erdos and Renyi shows that the random graph G(n,p)
experiences sharp phase transition around p=1/n - for any \epsilon>0 and
p=(1-\epsilon)/n, all connected components of G(n,p) are typically of size
O(log n), while for p=(1+\epsilon)/n, with high probability there exists a
connected component of size linear in n. We provide a very simple proof of this
fundamental result; in fact, we prove that in the supercritical regime
p=(1+\epsilon)/n, the random graph G(n,p) contains typically a path of linear
length. We also discuss applications of our technique to other random graph
models and to positional games.
