2,422 research outputs found
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
On sets with rank one in simple homogeneous structures
We study definable sets of SU-rank 1 in , where is a
countable homogeneous and simple structure in a language with finite relational
vocabulary. Each such can be seen as a `canonically embedded structure',
which inherits all relations on which are definable in , and has no
other definable relations. Our results imply that if no relation symbol of the
language of has arity higher than 2, then there is a close relationship
between triviality of dependence and being a reduct of a binary random
structure. Somewhat more preciely: (a) if for every , every -type
which is realized in is determined by its sub-2-types
, then the algebraic closure restricted to is
trivial; (b) if has trivial dependence, then is a reduct of a binary
random structure
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