# Some Themes Around First Order Theories Without the Independence Property

Chapter

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## Abstract

The aim of these notes (as well as the course of lectures they are based on) is to describe some current work around theories with *NIP* (not the independence property). This is a broad class of first order theories, including natural examples such as algebraically closed fields, differentially closed fields (both of which are stable) as well as real closed fields, *p*-adically closed fields and algebraically closed valued fields (which are unstable).This is really a paper on “pure” model theory, but I will comment here and there on applications and connections.

## Notes

### Acknowledgements

Supported by EPSRC grant EP/I002294/1.

## References

- 1.N. Alon, D.J. Kleitman, Piercing convex sets and the Hadwiger-Debrunner (
*p*,*q*)- problem. Adv. Math.**96**, 103–112 (1992)MathSciNetCrossRefzbMATHGoogle Scholar - 2.A. Chernikov, I. Kaplan, Forking and dividing in
*NTP*_{2}theories. J. Symb. Log.**77**, 1–20 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 3.A. Chernikov, P. Simon, Externally definable sets and dependent pairs. Isr. J. Math.
**194**, 409–425 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 4.A. Chernikov, P. Simon, Externally definable sets and dependent pairs II. Trans. AMS (to appear)Google Scholar
- 5.A. Chernikov, I. Kaplan, S. Shelah, On nonforking spectra (preprint, 2012)Google Scholar
- 6.E. Hrushovski, F. Loeser,
*Non-archimedean Tame Topology, and Stably Dominated Types*. Princeton Monograph Series (to appear)Google Scholar - 7.E. Hrushovski, Y. Peterzil, A. Pillay, Groups, measures and the
*NIP*. J. AMS**21**, 563–596 (2008).Google Scholar - 8.E. Hrushovski, A. Pillay, On NIP and invariant measures. J. Eur. Math. Soc.
**13**, 1005–1061 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 9.E. Hrushovski, A. Pillay, P. Simon, On generically stable and smooth measures in
*NIP*theories. Trans. AMS**365**, 2341–2366 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 10.H.K. Keisler, Measures and forking. Ann. Pure Appl. Log.
**45**, 119–169 (1987)MathSciNetCrossRefGoogle Scholar - 11.J. Matousek, Bounded VC-dimension implies a fractional Helly theorem. Discrete Comput. Geom.
**31**, 251–255 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 12.A. Pillay,
*Geometric Stability Theory*(Oxford University Press, Oxford, 1996)zbMATHGoogle Scholar - 13.A. Pillay, Externally definable sets and a theorem of Shelah, in
*Felgner Festchrift*(College Publications, London, 2007)Google Scholar - 14.A. Pillay, On weight and measure in
*NIP*theories. Notre Dame J. Formal Log. (to appear)Google Scholar - 15.S. Shelah, Dependent first order theories, continued. Isr. J. Math.
**173**, 1–60 (2009)CrossRefzbMATHGoogle Scholar - 16.S. Shelah, Dependent dreams and counting types (preprint 2012)Google Scholar
- 17.S. Shelah, Strongly dependent theories. Isr. J. Math. (to appear)Google Scholar
- 18.V. N. Vapnik, A.Y. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities. Theory Prob. Appl.
**16**, 264–280 (1971)CrossRefzbMATHGoogle Scholar

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