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STRONGLY MINIMAL STEINER SYSTEMS I: EXISTENCE

Part of: Model theory Finite geometry and special incidence structures

Published online by Cambridge University Press:  22 October 2020

JOHN BALDWIN  and
GIANLUCA PAOLINI
JOHN BALDWIN
Affiliation:
DEPARTMENT OF MATHEMATICS, STATISTICS, AND COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT CHICAGOCHICAGO, USAE-mail: [email protected]
GIANLUCA PAOLINI
Affiliation:
DEPARTMENT OF MATHEMATICS “GIUSEPPE PEANO” UNIVERSITY OF TORINO, VIA CARLO ALBERTO 10 10123, ITALYE-mail: [email protected]
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Abstract

A linear space is a system of points and lines such that any two distinct points determine a unique line; a Steiner k-system (for $k \geq 2$ ) is a linear space such that each line has size exactly k. Clearly, as a two-sorted structure, no linear space can be strongly minimal. We formulate linear spaces in a (bi-interpretable) vocabulary $\tau $ with a single ternary relation R. We prove that for every integer k there exist $2^{\aleph _0}$ -many integer valued functions $\mu $ such that each $\mu $ determines a distinct strongly minimal Steiner k-system $\mathcal {G}_\mu $ , whose algebraic closure geometry has all the properties of the ab initio Hrushovski construction. Thus each is a counterexample to the Zilber Trichotomy Conjecture.

Keywords

strongly minimalSteiner systemHrushovski construction

MSC classification

Primary: 03C45: Classification theory, stability and related concepts
Secondary: 51E10: Steiner systems
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 4 , December 2021 , pp. 1486 - 1507
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Copyright
© Association for Symbolic Logic 2020

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