In this article, we develop and clarify some of the basic combinatorial properties of the new notion of n-dependence 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 -partite -hypergraph with a definable edge relation. We characterize n-dependence by counting φ-types over finite sets, and in terms of the collapse of random ordered -hypergraph (...) indiscernibles down to order-indiscernibles. (shrink)

Building on Pierre Simon’s notion of distality, we introduce distality rank as a property of first-order theories and give examples for each rankmsuch that$1\leq m \leq \omega $. For NIP theories, we show that distality rank is invariant under base change. We also define a generalization of type orthogonality calledm-determinacy and show that theories of distality rankmrequire certain products to bem-determined. Furthermore, for NIP theories, this behavior characterizesm-distality. If we narrow the scope to stable theories, we observe thatm-distality can be (...) characterized by the maximum cycle size found in the forking “geometry,” so it coincides with$(m-1)$-triviality. On a broader scale, we see thatm-distality is a strengthening of Saharon Shelah’s notion ofm-dependence. (shrink)