In this talk we will prove that for a set S of tournaments the following
three statements are equivalent: 
- there exists k such that all members of S have pathwidth less than k; 
- there exists k such that no member of S contains k vertices that are pairwise k-connected; 
- there exists a digraph H such that no member of S contains a subdivision of H. 

As a consequence, we obtain a polynomial time algorithm to test whether a
tournament contains a subdivision of a fixed digraph H. We note that the
equivalent problem in general digraphs is NP-complete. 

Joint work with Paul Seymour.