Supersimple omega-categorical theories and pregeometries

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.