The Small index property for countable 1-transitive linear orders

It is shown that the countable saturated discrete linear ordering has the small index property, but that the countable 1-transitive linear orders which contain a convex subset isomorphic to ${\Bbb Z}^2$ do not. Similar results are also proved in the coloured case

Branchwise-real trees and bisimulations of potentialist systems

This thesis concerns two topics. The first treats R-trees, which are a certain kind of metric space tree in which every point can be branching. Favre and Jonsson posed the following problem in 2004: can the class of partial orders underlying R-trees be characterised by the fact that every branch is order-isomorphic to a real interval? I first answer this question in the negative, then go on to establish a connection between these trees and traditional set-theoretic trees. This connection is then put to work, answering refinements of Favre and Jonsson's question, yielding several independence results. I next move on to consider the existence of examples of these partial orders without non-trivial automorphisms. I provide constructions of these subject to increasingly strong uniformity conditions. While these constructions all take place in ZFC, they have a strong forcing flavour. The second topic deals with bisimulations of potentialist systems, which are first-order Kripke models based on embeddings. Given a first-order theory T we can impose a potentialist structure on the class of models of T by taking either all embeddings or all substructure inclusions between models. I show that these two systems are always bisimilar. Next, by connecting with a generalisation of the Ehrenfeucht-Fraïsé game, I show the equivalence of the existence of a bisimulation with elementary equivalence with respect to an infinitary language. Finally, I turn to the question of when a class-sized potentialist system is bisimilar to a set-sized one, providing two different sufficient conditions

On Ramsey properties of classes with forbidden trees

Let F be a set of relational trees and let Forbh(F) be the class of all structures that admit no homomorphism from any tree in F; all this happens over a fixed finite relational signature $\sigma$. There is a natural way to expand Forbh(F) by unary relations to an amalgamation class. This expanded class, enhanced with a linear ordering, has the Ramsey property.Comment: Keywords: forbidden substructure; amalgamation; Ramsey class; partite method v2: changed definition of expanded class; v3: final versio

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

The externally definable Ramsey property and fixed points on type spaces

We discuss the externally definable Ramsey property, a weakening of the Ramsey property for ultrahomogeneous structures, where the only colourings considered are those that are externally definable: that is, definable with parameters in an elementary extension. We show a number of basic results analogous to the classical Ramsey theory, and show that, for an ultrahomogeneous structure M, the externally definable Ramsey property is equivalent to the dynamical statement that, for each natural number n, every subflow of the space of n-types with parameters in M has a fixed point. We discuss a range of examples, including results regarding the lexicographic product of structures.Comment: 42 pages, 1 figur