Highly linked graphs

A graph with at least 2k vertices is said to bek-linked if, for any choices ₁,...,s _k ,t ₁,...,t _k of 2k distinct vertices there are vertex disjoint pathsP ₁,...,P _k withP _i joinings _i tot _i , 1ik. Recently Robertson and Seymour 16] showed that a graphG isk-linked provided its vertex connectivityk(G) exceeds . We show here thatk(G)22k will do.