11 research outputs found
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
Overgroups of the Automorphism Group of the Rado Graph
We are interested in overgroups of the automorphism group of the Rado graph.
One class of such overgroups is completely understood; this is the class of
reducts. In this article we tie recent work on various other natural
overgroups, in particular establishing group connections between them and the
reducts.Comment: 11 pages, 2 figure
Switching with more than two colours
AbstractThe operation of switching a finite graph was introduced by Seidel, in the study of strongly regular graphs. We may conveniently regard a graph as being a 2-colouring of a complete graph; then the extension to switching of an m-coloured complete graph is easy to define. However, the situation is very different. For m>2, all m-coloured graphs lie in the same switching class. However, there are still interesting things to say, especially in the infinite case.This paper presents the basic theory of switching with more than two colours. In the finite case, all graphs on a given set of vertices are equivalent under switching, and we determine the structure of the switching group and show that its extension by the symmetric group on the vertex set is primitive.In the infinite case, there is more than one switching class; we determine all those for which the group of switching automorphisms is the symmetric group. We also exhibit some other cases (including the random m-coloured complete graph) where the group of switching-automorphisms is highly transitive.Finally we consider briefly the case where not all switchings are allowed. For convenience, we suppose that there are three colours of which two may be switched. We show that, in the case of almost all finite random graphs, the analogue of the bijection between switching classes and two-graphs holds
Group actions on amorphous sets and reducts of coloured ransom graphs
EThOS - Electronic Theses Online ServiceGBUnited Kingdo
Limits of cubes
The celebrated Urysohn space is the completion of a countable universal homogeneous metric space which can itself be built as a direct limit of finite metric spaces. It is our purpose in this paper to give another example of a space constructed in this way, where the finite spaces are scaled cubes. The resulting countable space provides a context for a direct limit of finite symmetric groups with strictly diagonal embeddings, acting naturally on a module which additively is the “Nim field ” (the quadratic closure of the field of order 2). Its completion is familiar in another guise: it is the set of Lebesgue-measurable subsets of the unit interval modulo null sets. We describe the isometry groups of these spaces and some interesting subgroups, and give some generalisations and speculations. Key words: metric space; isometry group; completion; cube
Filters, topologies and groups from the random graph
We investigate the filter generated by vertex neighbourhoods in the countable random graph R, and two related topologies, with emphasis on their automorphism groups. These include a number of highly transitive groups containing Aut(R)