A Guide to Nip Theories

INTRODUCTION This text is an introduction to the study of NIP (or dependent) theories. It is meant to serve two purposes. The first is to present various aspects of NIP theories and give the reader sufficient background material to understand the current research in the area. The second is to advertise the use of honest definitions, in particular in establishing basic results, such as the so-called shrinking of indiscernibles. Thus although we claim no originality for the theorems presented here, a few proofs are new, mainly in chapters 3, 4 and 9. We have tried to give a horizontal exposition, covering different, sometimes unrelated topics at the expense of exhaustivity. Thus no particular subject is dealt with in depth and mainly low-level results are included. The choices made reflect our own interests and are certainly very subjective. In particular, we say very little about algebraic structures and concentrate on combinatorial aspects. Overall, the style is concise, but hopefully all details of the proofs are given. A small number of facts are left to the reader as exercises, but only once or twice are they used later in the text. The material included is based on the work of a number of model theo-rists. Credits are usually not given alongside each theorem, but are recorded at the end of the chapter along with pointers to additional topics. We have included almost no preliminaries about model theory, thus we assume some familiarity with basic notions, in particular concerning com-pactness, indiscernible sequences and ordinary imaginaries. Those prerequisites are exposed in various books such as that of Poizat [96], Marker [82], Hodges [57] or the recent book [116] by Tent and Ziegler. The material covered in a one-semester course on model theory should suffice. No familiarity with stability theory is required. History of the subject. In his early works on classification theory, Shelah structured the landscape of first order theories by drawing dividing lines defined by the presence or absence of different combinatorial configurations. The most important one is that of stability. In fact, for some twenty years, pure model theory did not venture much outside of stable theories. 7 8 1. Introduction Shelah discovered the independence property when studying the possible behaviors for the function relating the size of a subset to the number of types over it. The class of theories lacking the independence property, or NIP theories, was studied very little …