Supersimple omega-categorical theories and pregeometries

DOI10.1016/J.APAL.2019.102718arXiv1801.05748MaRDI QIDQ6296611

Vera Koponen

Publication date: 17 January 2018

Abstract: We prove that if $T$ is an $omega$-categorical supersimple theory with nontrivial dependence (given by forking), then there is a nontrivial regular 1-type over a finite set of reals which is realized by real elements; hence forking induces a nontrivial pregeometry on the solution set of this type and the pregeometry is definable (using only finitely many parameters). The assumption about $omega$-categoricity is necessary. This result is used to prove the following: If $V$ is a finite relational vocabulary with maximal arity 3 and $T$ is a supersimple $V$-theory with elimination of quantifiers, then $T$ has trivial dependence and finite SU-rank. This immediately gives the following strengthening of a previous result of the author: if $mathcalM$ is a ternary simple homogeneous structure with only finitely many constraints, then $Th(mathcalM)$ has trivial dependence and finite SU-rank.

Mathematics Subject Classification ID

Classification theory, stability, and related concepts in model theory (03C45) Models with special properties (saturated, rigid, etc.) (03C50) Quantifier elimination, model completeness, and related topics (03C10) Model theory of denumerable and separable structures (03C15) Categoricity and completeness of theories (03C35)

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