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Preprint Number 696
696. Ove Ahlman and Vera Koponen On sets with rank one in simple homogeneous structures E-mail: Submission date: 12 March 2014. 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. Mathematics Subject Classification: 03C15, 03C50, 03C45, 03C10 Keywords and phrases: |

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