39 research outputs found
Distal and non-distal NIP theories
We study one way in which stable phenomena can exist in an NIP theory. We
start by defining a notion of 'pure instability' that we call 'distality' in
which no such phenomenon occurs. O-minimal theories and the p-adics for example
are distal. Next, we try to understand what happens when distality fails. Given
a type p over a sufficiently saturated model, we extract, in some sense, the
stable part of p and define a notion of stable-independence which is implied by
non-forking and has bounded weight. As an application, we show that the
expansion of a model by traces of externally definable sets from some adequate
indiscernible sequence eliminates quantifiers
The definable (p, q)-theorem for distal theories
Answering a special case of a question of Chernikov and Simon, we show that any non-dividing formula over a model M in a distal NIP theory is a member of a consistent definable family, definable over M