Infinite combinatorics and Ramsey theory

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Infinite combinatorics and Ramsey theory We propose a research program on the structural and combinatorial properties of relational structures, in particular through infinite combinatorics and structural Ramsey theory. The study of properties preserved by partitioning a given structure is prevalent throughout Mathematics, and fundamentally the nature of our program. Two relational structures are said to be siblings if each embeds in the other. Bonato and Tardif conjectured that trees either have a single isomorphism class of siblings, or infinitely many (the tree alternative property). More recently, Thomassé formulated the related conjecture that any countable relational structure has either a single isomorphism class of siblings, countably many, or else continuum many. Together with Sauer and Pouzet, we verified the tree alternative property conjecture for scattered trees. We propose to apply these techniques to countable aleph_0-categorical relational structures, not only counting siblings but moreover describing the structural properties of these twins. A first goal is to complete the case of trees, and longer term verify Thomassé's conjecture in general. After Imrich et al showed that the distinguishing number of the Rado graph is two, we (with Nguyen Van The and Sauer) computed the distinguishing number of various other countable homogeneous structures, including graphs and posets. We showed that this number is in most cases two or infinite, and besides a few exceptions conjecture that this is so for all primitive homogeneous countable structures. This appears very much related to similar conjectures within the realm of infinite permutation groups, and we thus propose to look into this connection. We recently began to investigate homogeneous metric spaces. In particular, together with Bonato, Pawliuk and Sauer, we are investigating the case of homogeneous Urysohn spaces of a given spectrum. Previous work with Delhomme, Pouzet and Sauer studying indivisible metric spaces led us to consider homogeneous ultrametric spaces. Recently, we showed that for an ultrametric space to be homogeneous, it suffices that isometries defined on singletons extend, i.e. that the group of isometries acts transitively. Some related problems remain. In particular, for which spectrum V are ultrametric spaces with values in V and transitive automorphism group homogeneous? Another project extends some of our previous work in Ramsey theory, in particular precompact expansions of homogeneous structures, to Euclidean spaces and the conjecture that all spherical sets must be Ramsey. More recently, Leader-Russell-Walters, after proving that all Ramsey sets are subtransitive, conjectured that all transitive sets are Ramsey. One question arising is whether ordered spherical Euclidean spaces are Ramsey. We are working toward a counterexample among affinely dependent spaces. Another important target is the order property for affinely independent spaces.