By Pierre Simon

ISBN-10: 1107057752

ISBN-13: 9781107057753

The examine of NIP theories has acquired a lot cognizance from version theorists within the final decade, fuelled via functions to o-minimal buildings and valued fields. This booklet, the 1st to be written on NIP theories, is an creation to the topic that would attract an individual attracted to version concept: graduate scholars and researchers within the box, in addition to these in close by parts akin to combinatorics and algebraic geometry. with out residing on anybody specific subject, it covers all the simple notions and provides the reader the instruments had to pursue examine during this quarter. An attempt has been made in every one bankruptcy to provide a concise and stylish route to the most effects and to emphasize the main worthy rules. specific emphasis is wear sincere definitions, dealing with of indiscernible sequences and measures. The suitable fabric from different fields of arithmetic is made available to the truth seeker.

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**Example text**

Xn ; y) and not on the parameters b. By A being “big or small”, we mean that there is no cardinality restriction on A: it could be a small subset, a definable set, a type-definable set etc. 4. Let M = (R; ≤), seen as a model of DLO. Then M is (uniformly) stably embedded in U. More generally, a small set A is weakly stably embedded in U if and only if all types over A are definable. The theory T is stable if and only if any set A is weakly stably embedded if and only if any set A is stably embedded.

Note that for every i < , as |Sφ (Ai )| < , we have |Sφ (Ai ) \ Si | < and as each type from that set has less than extensions to a type over A, the cardinality of {p ∈ Sφ (A) : p Ai ∈ / Si } is less that . Summing over i < , we see that |Sφ (A) \ S | < . It follows that every type in Si has at least extensions to a type in S . Let S< = i< Si and S≤ = S< ∪ S . We define a linear order on S≤ in the following way. For p, q ∈ S≤ , if p ⊆ q (resp. q ⊆ p), we set p ≤ q (resp. q ≤ p). Otherwise, let i < be maximal such that p Si = q Si .

It follows that there is some i < such that |Sq ∩ Si | > . The induction hypothesis gives, for every type p ∈ Sq ∩ Si , a tuple (bp0 , . . , bpn−1 ). We can find two distinct types p1 , p2 ∈ Sq ∩ Si for which the corresponding tuples are the same, equal to some (b0 , . . , bn−1 ). Let bn ∈ Ai be such that p1 φ(x; bn ) and p2 ¬φ(x; bn ) (exchanging the roles of p1 and p2 if necessary). Then for every : n + 1 → {0, 1}, the partial type q(x) ∧ k≤n φ(x; bk ) (k) is consistent. This finishes the induction step, and the proof.

### A Guide to NIP Theories by Pierre Simon

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