mlq header will be provided by the publisher A note on stable sets, groups and theories with NIP Alf Onshuus ∗ and Ya’acov Peterzil ∗∗ Universidad de los Andes, Departemento de Matemáticas, Cra. 1 No 18A-10, Bogotá, Colombia University of Haifa, Department of Mathematics, Mount Carmel, Haifa 31905 ISRAEL Key words Independence property, stability, o-minimality, th-forking Subject classification 03C45, 03C64 Let M be an arbitrary structure. 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 holds for example when the structure M is interpretable in an o-minimal structure. More generally, an M-definable set X is called 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. Copyright line will be provided by the publisher 1 Introduction and definitions In this note we prove that in a definable group G, with a uniform finite bound on the U-rank of definable stable subsets, there is a maximal (up to finite index) normal stable subgroup H, such that G/H is purely unstable (see the definitions below and Theorem 2.1). We started to work in this article while studying structures which are interpretable in o-minimal theories. All of these satisfy our assumption on stable subsets. We later generalized the results to groups satisfying NIP with a bound on the U-rank of stable sets. It turned out, following a suggestion of the referee, that by modifying slightly our definition of a stable set, the NIP assumptions can be omitted. This is the reason why we examine two variations on the notion of a stable definable set. In the first section we review some definitions; in Section 2 we prove the main result; in Section 3 we discuss how rosy theories fit with the main result of the paper; in Section 4 we discuss weakly stable sets in theories with NIP and prove that they are stable. We then give some examples and characterize a large family of theories that satisfy the conditions required for our main theorem. Throughout this paper we work with a model M which is an elementary substructure some “monster” model C . Recall the following definitions. Definition 1.1. Let M be any structure and let φ(¯ x, ¯ y) be a formula in L(M ). • A formula φ(¯ x, ¯ y) is said to have the order property if there are infinite sequences 〈¯ a i 〉 i∈ω and 〈 ¯ b j 〉 j∈ω of tuples from M such that M | = φ(¯ a i , ¯ b j ) if and only if i ≤ j . A formula φ(¯ x, ¯ y) is stable if it does not have the order property. M is stable if no formula in a monster model of Th(M ) has the order property. • A formula φ(¯ x, ¯ y) is said to have the strict order property if there is an infinite sequence 〈¯ a i 〉 i∈ω of tuples such that N | = ∀ ¯ x(φ(¯ x, ¯ a i ) ⇒ φ(¯ x, ¯ a j )) if and only if i ≤ j . * email: onshuus@uniandes.edu.co ** email: kobi@math.haifa.ac.il Copyright line will be provided by the publisher