  Isoperimetric inequalities have been studied since antiquity, and in recent
decades they have been studied extensively on discrete objects, such as the
hypercube. An important special case of this problem involves bounding the size
of the shadow of a set system, and the basic question was solved by Kruskal (in
1963) and Katona (in 1968). In this paper we introduce the concept of the
shadow \d\G of a collection \G of ordered graphs, and prove the following,
simple-sounding statement: if n \in \N is sufficiently large, |V(G)| = n for
each G \in \G, and |\G| < n, then |\d \G| \ge |\G|. As a consequence, we
substantially strengthen a result of Balogh, Bollob\'as and Morris on
hereditary properties of ordered graphs: we show that if \P is such a property,
and |\P_k| < k for some sufficiently large k \in \N, then |\P_n| is decreasing
for k \le n < \infty.
