An improved linear edge bound for graph linkages

A graph is said to be k-linked if it has at least 2k vertices and for every sequence s₁,…,s_k,t₁,…,t_k of distinct vertices there exist disjoint paths P₁,…,P_k such that the ends of P_i are s_i and t_i. Bollobás and Thomason showed that if a simple graph G on n vertices is 2k-connected and G has at least 11kn edges, then G is k-linked. We give a relatively simple inductive proof of the stronger statement that 8kn edges and 2k-connectivity suffice, and then with more effort improve the edge bound to 5kn.