Résume | NIP -- "not having the independence property" -- is a constraint on the combinatorial behaviour of the definable sets in a given theory. Roughly: NIP means that there is no family of definable sets that induces on an infinite set X the family of all subsets of X. Shelah's Conjecture proposes that any complete NIP theory of fields is the theory of a separably closed, real closed, "henselian", or finite field. I will explain how we can refine this conjecture by specifying the complete theories that may appear in the "henselian" case. This is joint work with Franziska Jahnke. |