Erd\H os, Lov\'asz and Spencer showed in the late 1970s that the dimension of
the region of k-vertex graph profiles, i.e., the region of feasible densities
of k-vertex graphs in large graphs, is equal to the number of non-trivial
connected graphs with at most k vertices. We determine the dimension of the
region of k-vertex tournament profiles. Our result, which explores an
interesting connection to Lyndon words, yields that the dimension is much
larger than just the number of strongly connected tournaments, which would be
the answer expected as the analogy to the setting of graphs