Binary primitive homogeneous simple structures

TL;DR

This paper proves that countable, binary, primitive, homogeneous, simple structures have SU-rank 1, making them essentially random structures, and explores related properties of omega-categorical structures and equivalence relations.

Contribution

It establishes that binary, primitive, homogeneous, simple structures are of SU-rank 1 and characterizes parameter-definable equivalence relations in omega-categorical contexts.

Findings

SU-rank of such structures is 1

Binary, primitive, homogeneous, simple structures are random

Characterization of parameter-definable equivalence relations

Abstract

Suppose that M is countable, binary, primitive, homogeneous, and simple, and hence 1-based. We prove that the SU-rank of the complete theory of M is~1. It follows that M is a random structure. The conclusion that M is a random structure does not hold if the binarity condition is removed, as witnessed by the generic tetrahedron-free 3-hypergraph. However, to show that the generic tetrahedron-free 3-hypergraph is 1-based requires some work (it is known that it has the other properties) since this notion is defined in terms of imaginary elements. This is partly why we also characterize equivalence relations which are definable without parameters in the context of omega-categorical structures with degenerate algebraic closure. Another reason is that such characterizations may be useful in future research about simple (nonbinary) homogeneous structures.