76 research outputs found
The conjugacy problem for automorphism groups of countable homogeneous structures
We consider the conjugacy problem for the automorphism groups of a number of
countable homogeneous structures. In each case we find the precise complexity
of the conjugacy relation in the sense of Borel reducibility
Conjugacy for Homogeneous Ordered Graphs
We show that for any countable homogeneous ordered graph G, the conjugacy problem for automorphisms of G is Borel complete. In fact we establish that each such G satisfies a strong extension property called ABAP, which implies that the isomorphism relation on substructures of G is Borel reducible to the conjugacy relation on automorphisms of G
Finite -connected homogeneous graphs
A finite graph \G is said to be {\em -connected homogeneous}
if every isomorphism between any two isomorphic (connected) subgraphs of order
at most extends to an automorphism of the graph, where is a
group of automorphisms of the graph. In 1985, Cameron and Macpherson determined
all finite -homogeneous graphs. In this paper, we develop a method for
characterising -connected homogeneous graphs. It is shown that for a
finite -connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is
--transitive or G_v^{\G(v)} is of rank and \G has girth , and
that the class of finite -connected homogeneous graphs is closed under
taking normal quotients. This leads us to study graphs where is
quasiprimitive on . We determine the possible quasiprimitive types for
in this case and give new constructions of examples for some possible types
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
Vertex-primitive digraphs with large fixity
The relative fixity of a digraph is defined as the ratio between the
largest number of vertices fixed by a nontrivial automorphism of and
the number of vertices of . We characterize the vertex-primitive
digraphs whose relative fixity is at least , and we show that there are
only finitely many vertex-primitive digraphs of bounded out-valency and
relative fixity exceeding a positive constant.Comment: 16 page
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