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COMBINATORIAL BOUNDS IN DISTAL STRUCTURES

Part of: Model theory Graph theory

Published online by Cambridge University Press:  13 October 2023

AARON ANDERSON Open the ORCID record for AARON ANDERSON [Opens in a new window]
AARON ANDERSON*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES, 520 PORTOLA PLAZA LOS ANGELES, CA 90095, USA
*
E-mail: aaronanderson@math.ucla.edu
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Abstract

We provide polynomial upper bounds for the minimal sizes of distal cell decompositions in several kinds of distal structures, particularly weakly o-minimal and P-minimal structures. The bound in general weakly o-minimal structures generalizes the vertical cell decomposition for semialgebraic sets, and the bounds for vector spaces in both o-minimal and p-adic cases are tight. We apply these bounds to Zarankiewicz’s problem in distal structures.

Keywords

model theorycombinatoricsdistalo-minimalcutting lemmaZarankiewicz’s problem

MSC classification

Primary: 03C45: Classification theory, stability and related concepts
Secondary: 03C64: Model theory of ordered structures; o-minimality 05C35: Extremal problems

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Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 33
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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