On the capacity of digraphs

For a digraph G = (V, E) let w(Gⁿ) denote the maximum possible cardinality of a subset S of Vⁿ in which for every ordered pair (u₁, u₂, . . . , u_n) and (v₁, v₂, . . . , v_n) of members of S there is some 1 ≤ i ≤ n such that (u_i, v_i) ∈ E. The capacity C(G) of G is C(G) = lim_n→∞[(w(Gⁿ))^1/n]. It is shown that for any digraph G with maximum outdegree d, C(G) ≤ d + 1. It is also shown that for every n there is a tournament T on 2n vertices whose capacity is at least √n, whereas the maximum number of vertices in a transitive subtournament in it is only O(log n). This settles a question of Körner and Simonyi.