242 research outputs found
Reducts of structures and maximal-closed permutation groups
Answering a question of Junker and Ziegler, we construct a countable first order structure which is not ω-categorical, but does not have any proper nontrivial reducts, in either of two senses (model-theoretic, and group-theoretic). We also construct a strongly minimal set which is not ω-categorical but has no proper nontrivial reducts in the model-theoretic sense
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
All reducts of the random graph are model-complete
We study locally closed transformation monoids which contain the automorphism
group of the random graph. We show that such a transformation monoid is locally
generated by the permutations in the monoid, or contains a constant operation,
or contains an operation that maps the random graph injectively to an induced
subgraph which is a clique or an independent set. As a corollary, our
techniques yield a new proof of Simon Thomas' classification of the five closed
supergroups of the automorphism group of the random graph; our proof uses
different Ramsey-theoretic tools than the one given by Thomas, and is perhaps
more straightforward. Since the monoids under consideration are endomorphism
monoids of relational structures definable in the random graph, we are able to
draw several model-theoretic corollaries: One consequence of our result is that
all structures with a first-order definition in the random graph are
model-complete. Moreover, we obtain a classification of these structures up to
existential interdefinability.Comment: Technical report not intended for publication in a journal. Subsumed
by the more recent article 1003.4030. Length 14 pages
The Reducts of the Homogeneous Binary Branching C-relation
Let (L;C) be the (up to isomorphism unique) countable homogeneous structure
carrying a binary branching C-relation. We study the reducts of (L;C), i.e.,
the structures with domain L that are first-order definable in (L;C). We show
that up to existential interdefinability, there are finitely many such reducts.
This implies that there are finitely many reducts up to first-order
interdefinability, thus confirming a conjecture of Simon Thomas for the special
case of (L;C). We also study the endomorphism monoids of such reducts and show
that they fall into four categories.Comment: 39 pages, 4 figure
NIP omega-categorical structures: the rank 1 case
We classify primitive, rank 1, omega-categorical structures having
polynomially many types over finite sets. For a fixed number of 4-types, we
show that there are only finitely many such structures and that all are built
out of finitely many linear orders interacting in a restricted number of ways.
As an example of application, we deduce the classification of primitive
structures homogeneous in a language consisting of n linear orders as well as
all reducts of such structures.Comment: Substantial changes made to the presentation, especially in sections
3 and
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