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

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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.


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Author information

Correspondence to H. R. Johnson.

Additional information

M.C. Laskowski partially supported by NSF grant DMS-0600217.

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Johnson, H.R., Laskowski, M.C. Compression Schemes, Stable Definable Families, and o-Minimal Structures. Discrete Comput Geom 43, 914–926 (2010). https://doi.org/10.1007/s00454-009-9201-3

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  • Compression scheme
  • VC dimension
  • NIP
  • Independence dimension
  • Dependence
  • Warmuth conjecture
  • Stable
  • o-minimal
  • Type definition
  • Definable type
  • Density
  • Combinatorial complexity
  • UFTD