Relational Structures and Applications

本研究计划旨在进一步了解合并类和同构结构的friss<s:1>理论、Ramsey理论和自同构群的拓扑动力学之间的联系，以推动新的应用。我们建议作为一个团队正式加入我们的努力，作为这个计划的一部分，特别是提出以下研究方向：预紧展开目标是确定一个最优的齐次结构展开，以获得具有展开性质的Ramsey类，从而提供描述普遍最小流的存在性和方法。这里的方法是在少量精心选择的谓词的帮助下捕获基本的结构属性，这与该团队所熟知的规范分区的工作非常相关。著名的畸变问题问的是希尔伯特球是否近似不可分。在拓扑群中引入了振荡稳定性的概念，并证明这些概念在齐次度量空间中是等价的。随后得到了几个结果，包括证明了乌里松球振荡稳定性的一般结果。这是一个非常丰富的领域，深受分析问题的影响，但也有一些尚未解决的基本问题。我们探索了Rado图的自同构群的各种自然过群，推广了已知的约简方法。这项工作中令人惊讶的结果指出了进一步研究其他已知齐次结构的相应过群的工作，这些工作与可数齐次结构副本的超图的研究非常相关，提供了对哪些映射推广图同胚保存副本的见解。约束识别最近在复杂性问题的应用中涉及到同质结构、omega-categoricity和约束满足。我们建议调查一些非常自然的问题，一般研究- - -范畴结构和上述主题的减少。这个奇怪的概念是对称的一个粗略度量，我们已经推测原始结构应该有一个区分数，要么是2，要么是。这项工作似乎与无限置换群领域内的类似猜想非常相关，我们建议将这一参数的研究扩展到齐次结构。这一基本概念探讨了两个关系结构相互嵌入的条件，当人们研究固定年龄的关系结构的可嵌入性的预先顺序时自然产生。在嵌入的背景下，集的Cantor-Bernstein定理的失败是应用于线性顺序、图和竞赛研究的结果的关键。我们正在使用这些来自关系结构研究的工具，例如链性和核的概念，在这一领域取得进展，迄今为止，这一领域的研究主要集中在图和树上。C-XY同质结构这些是经典同质性概念的推广。C-HH结构的分类最近由D. Lockett完成，NSERC USRA的一个团队在有限C-MH情况下取得了实质性进展，为在可数情况下完成这类结构的分类提供了新的希望。我们已经开发了一些技术，包括包含区间的概念、区间分解和字典积，以理解原始结构和双赢图。我们将在无限的背景下进一步研究临界原性的概念。

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.