A note on stable sets, groups, and theories with NIP

Let M be an arbitrary structure. Then we say that an M ‐formula φ ( x ) defines a stable set in M if every formula φ ( x ) ∧ α ( x , y ) is stable. We prove: If G is an M ‐definable group and every definable stable subset of G has U ‐rank at most n (the same n for all sets), then G has a maximal connected stable normal subgroup H such that G / H is purely unstable. The assumptions hold for example if M is interpretable in an o‐minimal structure. More generally, an M ‐definable set X is weakly stable if the M ‐induced structure on X is stable. We observe that, by results of Shelah, every weakly stable set in theories with NIP is stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)