A high-dimensional approach to Ramsey Theory

拉姆齐理论的高维方法

• 批准号: EP/Y006399/1
• 负责人: Eoin Long
• 金额: $ 50.75万
• 依托单位: University of Birmingham
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY006399%2F1
• 关键词: dimensional; approach; Ramsey; Theory

Ramsey theory is a deep, influential and beautiful branch of Mathematics. The guiding philosophy here is that, in many situations, there is underlying order or predictability in large complex structures. A quick illustration is the fact that in any group of six people there will be three people who either (i) have all met one another or (ii) have all not met one another. The word `any' is important here -- such order is guaranteed to be present, regardless of the group. A similar, more general, statement also holds when `three' above is replaced by a larger number, provided the initial group is big enough.Order like this appears in a surprisingly wide range of contexts and is often extremely useful. As a result, Ramsey-type results have had significant impact across Mathematics, and beyond, including in Combinatorics, Ergodic Theory, Functional Analysis, Geometry, Mathematical Logic, Number Theory and Theoretical Computer Science. Ramsey theory has also proved to be particularly fertile ground for new ideas, and was instrumental in the development of several research areas and techniques, including Random Graph Theory, Regularity Methods, and the Probabilistic Method.Despite its impact and power, there are still fundamental aspects of Ramsey theory where we have surprisingly limited understanding. Historically, Ramsey theory for graphs, or networks, has best illustrated such challenges, and this proposal aims to resolve several significant and well-studied conjectures in this setting. The proposed research has two overarching goals. The first is to deepen our understanding of the structure of graphs which do not contain large homogeneous sets; roughly, these are pieces of the graph where it looks particularly simple. These graphs have a central role in Graph Ramsey Theory, but they remain quite mysterious. The second goal is to extend the applicability of powerful techniques for embedding graphs to much broader settings, which have up until now been out of reach.An important additional goal of this research is to develop new, flexible approaches to study these problems, combining tools from Extremal Set Theory, High-Dimensional Geometry and Probability to investigate the structure of graphs and hypergraphs. I have made novel use of the interplay of such tools in recent work, leading to the resolution of several well-known problems in Ramsey theory and Extremal Hypergraph Theory, but these ideas have much further potential. The research proposed below will greatly strengthen these connections, address the goals above, and provide new understanding in this important area.

拉姆齐理论是数学中一个博大精深、影响深远而又美丽的分支。这里的指导思想是，在许多情况下，在大型复杂结构中存在潜在的秩序或可预测性。一个简单的例子是，在任何六个人组成的小组中，将有三个人要么(I)都见过面，要么(Ii)都没有见过面。“任何”这个词在这里很重要--这样的秩序肯定会存在，无论是哪个群体。类似的，更一般的说法也适用，当上面的‘三’被一个更大的数字代替时，只要最初的组足够大。这样的顺序出现在令人惊讶的广泛的上下文中，而且通常非常有用。因此，Ramsey类型的结果对数学产生了重大影响，包括组合学、遍历理论、泛函分析、几何、数理逻辑、数论和理论计算机科学。拉姆齐理论也被证明是产生新思想的特别肥沃的土壤，并在几个研究领域和技术的发展中发挥了重要作用，包括随机图理论、正则性方法和概率方法。尽管拉姆齐理论具有影响力和威力，但我们对拉姆齐理论的基本方面的理解仍然令人惊讶地有限。从历史上看，拉姆齐关于图或网络的理论最好地说明了这种挑战，这一建议旨在解决这种背景下几个重要的和经过充分研究的猜想。这项拟议的研究有两个首要目标。首先是加深我们对不包含大的齐次集的图的结构的理解；粗略地说，这些是图的部分，在那里它看起来特别简单。这些图在拉姆齐图理论中起着核心作用，但它们仍然相当神秘。第二个目标是将强大的图嵌入技术的适用性扩展到更广泛的环境中，这一研究的另一个重要目标是开发新的、灵活的方法来研究这些问题，结合极值集合论、高维几何和概率来研究图和超图的结构。在最近的工作中，我新颖地使用了这些工具的相互作用，导致了Ramsey理论和极值超图理论中几个著名问题的解决，但这些想法还有更大的潜力。下面提出的研究将极大地加强这些联系，解决上述目标，并在这一重要领域提供新的理解。