Disjoint paths in graphs

Suppose that (s₁, t₁),...,(s_k, t_k) are pairs of vertices of a graph. When can one choose a path between s₁ and t₁ for each i, all pairwise edge-disjoint? Menger's theorem answers this when s₁,...,s_k, t₁,...,t_k take only two distinct values, but the general problem is unsolved. We settle the two next simplest cases. 1. (i) when k = 2, and 2. (ii) when s₁,...,s_k, t₁,...,t_k take only three distinct values-the solution to this is obtained by applying a theorem of Mader. We obtain both good characterizations and good algorithms for these problems. The analogous "vertex-disjoint" problems are also solved.