Supersimple omega-categorical theories and pregeometries

TL;DR

This paper demonstrates that in omega-categorical supersimple theories with nontrivial dependence, a definable pregeometry arises, and it establishes conditions under which such theories have trivial dependence and finite SU-rank, especially in finite relational vocabularies.

Contribution

It proves the existence of a nontrivial definable pregeometry in omega-categorical supersimple theories with dependence, and shows that certain finite relational theories have trivial dependence and finite SU-rank.

Findings

Existence of a nontrivial regular 1-type with a definable pregeometry.

Supersimple theories with elimination of quantifiers in vocabularies of arity 3 have trivial dependence.

Ternary simple homogeneous structures with finitely many constraints have trivial dependence and finite SU-rank.

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 $\mathcal{M}$ is a ternary simple homogeneous structure with only finitely many constraints, then $Th(\mathcal{M})$ has trivial dependence and finite SU-rank.