  In this note we develop and clarify some of the basic combinatorial
properties of the new notion of $n$-dependence (for $1\leq n < \omega$)
recently introduced by Shelah. In the same way as dependence of a theory means
its inability to encode a bipartite random graph with a definable edge
relation, $n$-dependence corresponds to the inability to encode a random
$(n+1)$-partite $(n+1)$-hypergraph with a definable edge relation. Most
importantly, we characterize $n$-dependence by counting $\varphi$-types over
finite sets (generalizing Sauer-Shelah lemma and answering a question of
Shelah) and in terms of the collapse of random ordered $(n+1)$-hypergraph
indiscernibles down to order-indiscernibles (which implies that the failure of
$n$-dependence is always witnessed by a formula in a single free variable).
