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Binary primitive homogeneous simple structures
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.Comment: In this revised version of the article, the title has been changed
from 'Binary primitive homogeneous 1-based structures' to 'Binary primitive
homogeneous simple structures', because it was found that every binary,
homogeneous, simple structure is 1-base