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… (More)
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).
Let R m be the (unique) universal homogeneous m-edge-coloured countable complete graph (m ≥ 2), and G m its group of colour-preserving automorphisms. The group G m was shown to be simple by John Truss. We examine the automorphism group of G m , and show that it is the group of permutations of R m which induce permutations on the colours, and hence an… (More)
The 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… (More)