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By Pierre Simon

The learn of NIP theories has bought a lot consciousness from version theorists within the final decade, fuelled through purposes to o-minimal buildings and valued fields. This publication, the 1st to be written on NIP theories, is an advent to the topic that may entice a person drawn to version conception: graduate scholars and researchers within the box, in addition to these in close by components equivalent to combinatorics and algebraic geometry. with out living on anyone specific subject, it covers all the easy notions and offers the reader the instruments had to pursue study during this region. An attempt has been made in each one bankruptcy to provide a concise and stylish route to the most effects and to emphasize the main invaluable rules. specific emphasis is wear sincere definitions, dealing with of indiscernible sequences and measures. The appropriate fabric from different fields of arithmetic is made available to the truth seeker.

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3. Pairs. Another way to single out a subset A from U is to name the set A by a new unary predicate. This creates what we will call a pair. More precisely, let M be an L structure, A ⊆ M . Let LP = L ∪ {P(x)}, where P(x) is a new unary relation symbol. We will sometimes write x ∈ P instead of P(x). The pair (M, A) is the LP -structure whose L-reduct is M , and where P is interpreted by: P(a) ⇐⇒ a ∈ A. The structure (M, A) is richer than M and Aind(M ) in the sense that both are interpretable in it.

Therefore aktcc |= ptc A ∪ {aktmm : m = c}. 6. Let (It : t ∈ X ) be a family of mutually indiscernible sequences. For t ∈ X , let pt = lim(It ). , pt (x) ⊗ ps (y) = ps (y) ⊗ pt (x)). 7. Let (It : t ∈ X ) be a family of sequences, with It = (ait : i ∈ It ). Assume that the sequences (It : t ∈ X ) are mutually indiscernible over A. Let J be a linearly ordered set, and for t ∈ X , let t : J → It be an increasing embedding. Then the sequence ((a t t (i) )t∈X : i ∈ J) is indiscernible over A. 8. Let (It : t ∈ X ) be a family of sequences mutually indiscernible over A.

12. Let I ⊂ M be an indiscernible sequence. Let (M , I ) be an elementary extension of the pair (M, I ). Then there is an ordering on I making it into an indiscernible sequence. 2. Honest definitions We now state the main theorem of this chapter. 13. Let M |= T , A ⊆ M , φ(x; y) ∈ L and b ∈ M a |y|tuple. Assume that φ(x; y) is NIP. Then there is an elementary extension (M, A) ≺ (M , A ), a formula (x; z) ∈ L and a tuple d of elements of A such that φ(A; b) ⊆ (A ; d ) ⊆ φ(A ; b). 14. The conclusion of the theorem is equivalent to saying that there is a formula (x; z) such that for any finite A0 ⊆ φ(A; b), there is a tuple d of elements of A such that A0 ⊆ (A; d ) ⊆ φ(A; b).

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