The extremal function for 3-linked graphs

A graph is k-linked if for every set of 2k distinct vertices { s 1 , … , s k , t 1 , … , t k } there exist disjoint paths P 1 , … , P k such that the endpoints of P i are s i and t i . We prove every 6-connected graph on n vertices with 5 n − 14 edges is 3-linked. This is optimal, in that there exist 6-connected graphs on n vertices with 5 n − 15 edges that are not 3-linked for arbitrarily large values of n.