On sets with rank one in simple homogeneous structures

TL;DR

This paper investigates definable sets of SU-rank 1 in countable homogeneous simple structures, revealing their connection to binary random structures under certain conditions, and exploring the triviality of dependence.

Contribution

It establishes a link between trivial dependence and reducts of binary random structures for definable sets of SU-rank 1 in simple homogeneous structures.

Findings

If n-types are determined by 2-types, then algebraic closure on D is trivial.

If M has trivial dependence, then D is a reduct of a binary random structure.

Abstract

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 preciely: (a) if for every $n \geq 2$, every $n$-type $p(x_1, ..., x_n)$ which is realized in $D$ is determined by its sub-2-types $q(x_i, x_j) \subseteq p$, 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.