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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.

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