  For a first-order formula $\phi(x;y)$ we introduce and study the
characteristic sequence $<P_n : n < \omega>$ of hypergraphs defined by
$P_n(y_1,...,y_n) := (\exists x) \bigwedge_{i \leq n} \phi(x;y_i)$. We show
that combinatorial and classification theoretic properties of the
characteristic sequence reflect classification theoretic properties of $\phi$
and vice versa. Specifically, we show that some tree properties are detected by
the presence of certain combinatorial configurations in the characteristic
sequence while other properties such as instability and the independence
property manifest themselves in the persistence of complicated configurations
under localization.
