Infinite combinatorics and Ramsey theory

我们提出了一个关于关系结构的结构和组合性质的研究计划，特别是通过无限组合学和结构拉姆齐理论。通过分割给定结构来研究保持的性质在整个数学中很普遍，并且从根本上说是我们程序的性质。如果两个关系结构中的一个嵌入另一个，那么这两个关系结构被称为兄弟关系。Bonato和Tardif指出树要么有一个同构类的兄弟，要么有无穷多个（树的替代性质）。更近一些时候，卡萨塞提出了一个相关的猜想，即任何可数关系结构要么有一个单一的同构类的兄弟姐妹，可数多，要么连续多。与Sauer和Pouzet一起，我们验证了树的择一性猜想.我们建议将这些技术应用于可数的aleph_0-范畴关系结构，不仅计算兄弟姐妹，而且描述这些双胞胎的结构特性。第一个目标是完成树的情况下，并长期验证一般的Mallassé猜想。 在Imrich等人证明Rado图的可区别数为2之后，我们（与Nguyen货车The和Sauer）计算了各种其他可数齐次结构的可区别数，包括图和偏序集。我们证明了这个数在大多数情况下是2或无穷大，除了少数例外，推测这是所有原始齐次可数结构。这似乎与无限置换群领域内的类似构造非常相关，因此我们建议研究这种联系。我们最近开始研究齐性度量空间。特别是，连同博纳托，Pawliuk和绍尔，我们正在调查的情况下，齐性Urysohn空间的一个给定的频谱。 以前的工作与Delhomme，Pouzet和绍尔研究不可分度量空间使我们考虑齐次超度量空间。最近，我们证明了，对于一个超度量空间是齐次的，定义在单态上的等距扩张就足够了，即等距群的作用是传递的。一些相关的问题仍然存在。特别地，对于哪些谱V是值在V中且传递自同构群齐次的超度量空间？另一个项目扩展了我们以前的一些工作在拉姆齐理论，特别是准紧扩张的齐次结构，欧几里德空间和猜想，所有的球面集必须拉姆齐。最近，Leader-Russell-Walters在证明了所有Ramsey集都是次传递集之后，又证明了所有传递集都是Ramsey集。由此产生的一个问题是有序球面欧氏空间是否是拉姆齐。我们正在研究仿射依赖空间中的一个反例。另一个重要的目标是仿射独立空间的序性质。

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.