Before revision:  Our results include the following: For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized O(n^{4/5})-competitive algorithm for graphs with general lengths, where n is the number of vertices of the given graph. For graphs with uniform lengths, we give an efficient randomized O(n^{2/3+\epsilon})-competitive algorithm, and an efficient deterministic O(k^{1/2+\epsilon})-competitive algorithm, where k is the number of terminal pairs.
After revision:  Our results include the following: For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized O(n^{4/5})-competitive algorithm for graphs with general lengths, where n is the number of vertices . With uniform lengths, we give an efficient randomized O(n^{2/3+\epsilon})-competitive algorithm, and an efficient deterministic O(k^{1/2+\epsilon})-competitive algorithm, where k is the number of terminal pairs.