Highly linked tournaments

A (possibly directed) graph is k  -linked if for any two disjoint sets of vertices {x1,…,xk}{x1,…,xk} and {y1,…,yk}{y1,…,yk} there are vertex disjoint paths P1,…,PkP1,…,Pk such that PiPi goes from xixi to yiyi. A theorem of Bollobás and Thomason says that every 22k-connected (undirected) graph is k  -linked. It is desirable to obtain analogues for directed graphs as well. Although Thomassen showed that the Bollobás–Thomason Theorem does not hold for general directed graphs, he proved an analogue of the theorem for tournaments—there is a function f(k)f(k) such that every strongly f(k)f(k)-connected tournament is k  -linked. The bound on f(k)f(k) was reduced to O(klog⁡k)O(klog⁡k) by Kühn, Lapinskas, Osthus, and Patel, who also conjectured that a linear bound should hold. We prove this conjecture, by showing that every strongly 452k-connected tournament is k-linked.