 
Summary: On the Capacity of Digraphs
Noga Alon
Abstract
For a digraph G = (V, E) let w(Gn
) denote the maximum possible cardinality of a subset S of
V n
in which for every ordered pair (u1, u2, . . . , un) and (v1, v2, . . . , vn) of members of S there is
some 1 i n such that (ui, vi) E. The capacity C(G) of G is C(G) = limn[ (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 KĻorner and Simonyi.
1 Introduction
For a digraph G = (V, E) and for a positive integer n, let w(Gn) denote the maximum possible
cardinality of a subset S of V n in which for every ordered pair (u1, u2, . . . , un) and (v1, v2, . . . , vn) of
members of S there is some i, 1 i n such that (ui, vi) is a directed edge of G. It is easy to see
