On sets with rank one in simple homogeneous structures
Volume 228 / 2015
Abstract
We study definable sets $D$ of SU-rank 1 in $\mathcal M^{\rm eq}$, where $\mathcal M$ 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 $\mathcal M^{\rm eq}$, and has no other definable relations. Our results imply that if no relation symbol of the language of $\mathcal M$ has arity higher than 2, then there is a close relationship between triviality of dependence and $\mathcal D$ being a reduct of a binary random structure. Somewhat more precisely: (a) if for every $n \geq 2$, every $n$-type $p(x_1, \ldots , x_n)$ which is realized in $D$ is determined by its sub-2-types $q(x_i, x_j) \subseteq p$, then the algebraic closure restricted to $D$ is trivial; (b) if $\mathcal M$ has trivial dependence, then $\mathcal D$ is a reduct of a binary random structure.
