 
Summary: VAPNIKCHERVONENKIS DENSITY IN SOME THEORIES
WITHOUT THE INDEPENDENCE PROPERTY, I
MATTHIAS ASCHENBRENNER, ALF DOLICH, DEIRDRE HASKELL,
DUGALD MACPHERSON, AND SERGEI STARCHENKO
For Lou van den Dries, on his 60th birthday.
Abstract. We recast the problem of calculating VapnikChervonenkis (VC) den
sity into one of counting types, and thereby calculate bounds (often optimal) on the
VC density for some weakly ominimal, weakly quasiominimal, and Pminimal
theories.
Contents
1. Introduction 1
2. VC Density 7
3. The ModelTheoretic Context 15
4. Some VC Density Calculations 27
5. Theories with the VC d Property 35
6. Examples of VC d: Weakly Ominimal Theories and Variants 42
7. A Strengthening of VC d, and Padic Examples 47
References 56
1. Introduction
The notion of VC dimension, which arose in probability theory in the work of Vapnik and
