Abstract
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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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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DOI: https://doi.org/10.1007/s00454-009-9201-3