### On non-forking spectra

by Chernikov and Kaplan and Shelah. [CeKpSh:1007]

J European Math Soc, 2016

Non-forking is one of the most important notions in modern model
theory capturing the idea of a generic extension of a type (which
is a
far-reaching generalization of the concept of a generic point
of a
variety).
To a countable first-order theory we associate its emph {non-forking
spectrum} --- a function of two cardinals kappa and lambda
giving the supremum of the possible number of types over a model
of
size lambda that do not fork over a sub-model of size kappa .
This is a natural generalization of the stability function of
a theory.
We make progress towards classifying the non-forking spectra. On the
one hand, we show that the possible values a non-forking spectrum
may take are quite limited. On the other hand, we develop a general
technique for constructing theories with a prescribed non-forking
spectrum, thus giving a number of examples. In particular, we answer
negatively a question of Adler whether NIP is equivalent to bounded
non-forking.
In addition, we answer a question of Keisler regarding the number
of cuts a linear order may have. Namely, we show that it is possible
that {ded} kappa <({ded} kappa)^{omega} .

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