Infinite combinatorics and Ramsey theory

无限组合学和Ramsey理论我们提出了一个关于关系结构的结构和组合性质的研究方案，特别是通过无限组合学和结构Ramsey理论。对通过划分给定结构保存的性质的研究在整个数学课程中很普遍，基本上也是我们程序的本质。如果两个关系结构相互嵌入，则称其为兄弟关系。Bonato和Tardif猜想，树要么只有一个同构的兄弟姐妹类，要么有无限多个(树的替代性质)。最近，托马塞提出了相关的猜想，即任何可数关系结构要么有单个同构类的兄弟，可数个，要么有连续的多个。我们与Sauer和Pouzet一起验证了散布树的树替代性质猜想。我们建议将这些技术应用于可数Aleph0范畴关系结构，不仅计算兄弟姐妹，而且描述这些孪生兄弟的结构性质。第一个目标是完成树的情况，并从更长期的角度验证托马塞的S猜想。在Imrich等人证明了Rado图的区别数为2之后，我们(与Nguyen Van the和Sauer)计算了其他各种可数齐次结构的区别数，包括图和偏序集。我们证明了这个数在大多数情况下是2或无限的，除了少数例外，我们猜想对于所有的本原齐次可数结构都是如此。这似乎与无限置换群领域内的类似猜想非常相关，因此我们建议研究这一联系。我们最近开始研究齐次度量空间。特别地，我们与Bonato，Pawliuk和Sauer一起研究了给定谱的齐次Urysohn空间的情形。以前与Delhomme，Pouzet和Sauer一起研究不可分度量空间的工作使我们考虑齐次超度量空间。最近，我们证明了对于齐次超距离空间，定义在单元上的等距扩张，即等距群起传递作用就足够了。一些相关的问题仍然存在。特别地，对于哪些谱V是取值于V且传递自同构群齐次的超度量空间？另一个项目将我们以前在Ramsey理论中的一些工作，特别是齐次结构的预紧展开，推广到欧几里得空间和所有球面集一定是Ramsey的猜想。最近，Leader-Russell-Walters在证明了所有Ramsey集都是次传递集之后，猜想所有传递集都是Ramsey集。由此产生的一个问题是有序球面欧几里德空间是否是Ramsey空间。我们正在努力在相互依附的空间中树立一个反例。另一个重要的目标是仿射独立空间的序性。

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.