Set theoretic classification results for relational structures

The unifying theme of my work involves relational structures. In particular, I study structures which have a single binary relation, such as graphs, linear orders, Boolean algebras and trees. Relational structures are ubiquitous in all areas of mathematics, computer and information sciences. The methods that I use are set-theoretic, mainly infinite combinatorics and forcing. My work has direct applications in measure and probability theory, combinatorics and model theory. The bulk of my work is to classify certain relational structures using their combinatorial properties or through the universality programme, which heavily relies on combinatorial methods. I concentrate on three sub- projects to this line of research. The first is to determine and find connections between universality spectra for non-elementary relational structures. Universal models are not only important for classification, but are important and well-studied structures in their own right. There is a strong programme in universality and much progress has been made on using this indicator to classify elementary structures in a model-theoretic way. However, non-elementary structures are not model- theoretically well-behaved and in my project we rely on set-theoretic methods to decide these questions. The second is to classify orders (linear and partial) which have generalised notions of dense and scattered. These classifications take the form of a constructive hierarchy. There are some very interesting open problems of this type that I plan to work on. For scattered orders, these constructive hierarchies proved to be a very useful tool for proving structure and combinatorial theorems about such orders and I plan to extend these results to orders which are scattered in a stronger sense. The third is to find purely combinatorial classifications of Boolean algebras which carry specific types of measures, including probability measures. The measures that I will consider are finitely-additive measures with different properties. These classifications will also be used to obtain a structure theory for such Boolean algebras. As part of the inner model programme, I am involved in finding internal consistency results for global properties. The properties that we consider come from my work in relational structures. This provides a rich base of examples to examine in order to prove more general theorems about which results can be true in an inner model. In my previous work I have found deep interconnections between different types of relational structures and between the concepts used to classify them. These connections have proved to be useful in finding new approaches to solve the problems above. In the classification projects that I plan to engage in, I will build on this foundation in order to discover and apply new connections.