Tournament pathwidth and topological containment

We prove that if a tournament has pathwidth ⩾4θ2+7θ then it has θ vertices that are pairwise θ-connected. As a consequence of this and previous results, we obtain that for every set S of tournaments the following are equivalent:
• there exists k such that every member of S has pathwidth at most k,
• there is a digraph H such that no subdivision of H is a subdigraph of any member of S,
• there exists k such that for each T∈S, there do not exist k vertices of T that are pairwise k-connected.As a further consequence, we obtain a polynomial-time algorithm to test whether a tournament contains a subdivision of a fixed digraph H as a subdigraph.