Compression Schemes, Stable Definable Families, and o-Minimal Structures

@article{Johnson2010CompressionSS,
  title={Compression Schemes, Stable Definable Families, and o-Minimal Structures},
  author={Hunter R. Johnson and Michael C. Laskowski},
  journal={Discrete \& Computational Geometry},
  year={2010},
  volume={43},
  pages={914-926}
}
We show that any family of sets uniformly definable in an o-minimal structure has an extended compression scheme of size equal to the number of parameters in the defining formula.As a consequence, the combinatorial complexity (or density) of any definable family in a structure with a o-minimal theory is bounded by the number of parameters in the defining formula.Extended compression schemes for uniformly definable families corresponding to stable formulas are also shown to exist. 
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