Submission date: 17 January 2018

Abstract:

We prove that if T is an ω-categorical supersimple theory with nontrivial dependence (given by dividing/forking), then there is a nontrivial regular 1-type over a finite set of reals which is realized by real elements; hence dividing induces a nontrivial pregeometry on the solution set of this type and the pregeometry is definable (using only finitely many parameters). The assumption about ω-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 M is a ternary simple homogeneous structure with only finitely many constraints, then Th(M) has trivial dependence and finite SU-rank.

Mathematics Subject Classification: 03C45, 03C10, 03C35, 03C68

Keywords and phrases:

Full text arXiv 1801.05748: pdf, ps.