On sets with rank one in simple homogeneous structures

摘要

We study definable sets D of SU-rank 1 in M-eq, where M is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such D can be seen as a 'canonically embedded structure', which inherits all relations on D which are definable in M-eq, and has no other definable relations. Our results imply that if no relation symbol of the language of M has arity higher than 2, then there is a close relationship between triviality of dependence and D being a reduct of a binary random structure. Somewhat more precisely: (a) if for every n >= 2, every n -type p (x(1),...,x(n)) which is realized in D is determined by its sub-2-types q (x(i), x(j)) subset of p, then the algebraic closure restricted to D is trivial; (b) if M has trivial dependence, then D is a reduct of a binary random structure.