On the Capacity of Digraphs

For a digraphG = (V,E) letw(Gn) denote the maximum possible cardinality of a subsetSofVnin which for every ordered pair (u1,u2, … ,un) and (v1,v2, … ,vn) of members ofSthere is some 1 ≤ i ≤ nsuch that (ui,vi) ∈ EThecapacityC(G) ofGisC(G) = limn↦∞[(w(Gn))1/n]. It is shown that for any digraphGwith maximum outdegreed,C(G) ≤ d + 1. It is also shown that for everynthere is a tournamentTon 2nvertices whose capacity is at leastn, whereas the maximum number of vertices in a transitive subtournament in it is onlyO(log n). This settles a question of Körner and Simonyi.