Infinite combinatorics and Ramsey theory

我们提出了一个研究关系结构的结构和组合性质的计划，特别是通过无限组合和结构拉姆齐理论。通过划分给定结构来保存属性的研究在整个数学中都很普遍，从根本上说，这也是我们程序的本质。******如果两个关系结构相互嵌入，则称为兄弟关系结构。Bonato和Tardif推测，树要么有一个同构的兄弟姐妹类，要么有无限多个（树的可选性）。最近，thomass<s:1>提出了一个相关的猜想，即任何可数关系结构要么有一个同构的兄弟姐妹类，可数多，要么连续多。我们与Sauer和Pouzet一起验证了散树的树可选性猜想。我们建议将这些技术应用于可数的alph_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，并从更长远的角度总体上验证托马斯的猜想。******在Imrich等人证明了Rado图的区分数是2之后，我们（与Nguyen Van the和Sauer）计算了其他各种可计数同构结构的区分数，包括图和poset。我们证明了这个数在大多数情况下是2或无限的，并且除了少数例外，我们推测所有原始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密切相关，因此我们建议研究这种联系。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空间的情况。******先前与Delhomme， Pouzet和Sauer一起研究不可分度量空间的工作使我们考虑了齐次超度量空间。最近，我们证明了对于一个超度量空间是齐次的，在单态上定义的等距是扩展的，即等距群是传递的。一些相关问题依然存在。特别是哪个谱V是值在V内的超度量空间和传递自同构群齐次的？******另一个项目将我们之前在拉姆齐理论中的一些工作，特别是齐次结构的预紧展开，扩展到欧几里德空间和所有球面集必须是拉姆齐的猜想。最近，Leader-Russell-Walters在证明了所有的Ramsey集合都是次传递的之后，又推测所有的传递集合都是Ramsey。一个问题是有序球面欧几里得空间是否是拉姆齐空间。我们正在研究仿射依赖空间的反例。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. **