Relational Structures and Applications

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This research program aims at further understanding the connections between the Fraissé theory of amalgamation classes and homogeneous structures, Ramsey theory, and topological dynamics of automorphism groups, driven toward new applications. We propose to formally join our efforts as a team as part of this program, in particular proposing the following lines of research:**Precompact expansions*The objective is to determine an optimal expansion of homogeneous structures to obtain a Ramsey class with the expansion property, thus providing the existence and means to describe the universal minimal flows. The approach here is to capture basic structural properties with the help of a minimal number of well chosen predicates, very much related to the work on canonical partitions well known by this team. **Oscillation stability *The famous distortion problem asked whether the Hilbert sphere is approximately indivisible. The notion of oscillation stability was introduced for topological groups, and it turns out that these concepts are equivalent for homogeneous metric spaces. Several results followed, including the proof of a general result that yields the oscillation stability of the Urysohn sphere. This is a very rich area very much influenced by problems in analysis, but with some unresolved fundamental questions.**Overgroups of automorphism groups*We have explored various natural overgroups of the automorphism group of the Rado graph, generalizing well known methods for reducts. The surprising results in this work point to further work investigating the corresponding overgroups of the other well known homogeneous structures, work very much related to the study of the hypergraph of copies of countable homogeneous structures providing insight into which maps generalizing graph homeomorphisms preserve the copies.**Constraint Recognition*There are recent connections involving homogeneous structures, omega-categoricity and constraint satisfaction with applications to complexity problems. We propose to investigate some very natural questions general to the study of reducts of omega-categorical structures and the themes above.**Distinguishing number*This curious notion is a rough measure of symmetry, and we have conjectured that primitive structures should have a distinguishing number of either 2 or omega. This work appears very much related to similar conjectures within the realm of infinite permutation groups, and we propose to extend the investigation of this parameter for homogeneous structures. **Equimorphy*This fundamental notion explores conditions when two relational structures embed in the other, arising naturally when one studies the preorder of embeddability on relational structures of a fixed age. The failure of the Cantor-Bernstein theorem for sets in the context of embeddings is key to results that are applied in the study of linear orders, graphs and tournaments. We are using these tools from the study of relational structures, such as the notion of chainability and kernel to make advances in this area which has to date largely been studied only for graphs and trees.**C-XY homogeneous structures*These generalize the classical homogeneity notion. The classification of the C-HH structures was recently done by D. Lockett, and a team's NSERC USRA has made substantial progress in the finite C-MH case, providing renewed hope for the complete classification of this class in the countable case. **Intervals in relational structures*We have developed techniques, including the notion of inclusive intervals, interval decomposition and lexicographic products, toward understanding primitive structures and cop-win graphs. We will further investigate notions of critical primitivity in the infinite context.