A high-dimensional approach to Ramsey Theory

Ramsey理论是数学中一个深刻的、有影响的、美丽的分支。这里的指导思想是，在许多情况下，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，在任何一个六人的小组中，都会有三个人（i）彼此都见过面，或者（ii）彼此都没有见过面。“任何”一词在这里很重要-这种秩序是保证存在的，而不管是哪一组。当前面的“3”被一个更大的数所取代时，一个类似的、更一般的陈述也成立，只要最初的一组足够大。因此，拉姆齐类型的结果在数学领域产生了重大影响，包括组合数学，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新思想的沃土，并在几个研究领域和技术的发展中发挥了重要作用，包括随机图论，正则性方法和概率方法。尽管拉姆齐理论的影响和力量，我们仍然对拉姆齐理论的基本方面有着惊人的有限理解。从历史上看，拉姆齐理论的图，或网络，最好地说明了这样的挑战，这个建议的目的是解决几个重要的和良好的研究在这种情况下。拟议的研究有两个总体目标。首先是加深我们对不包含大型齐次集的图的结构的理解;粗略地说，这些是图中看起来特别简单的部分。这些图在图拉姆齐理论中扮演着核心角色，但它们仍然非常神秘。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，这是迄今为止无法实现的。本研究的一个重要的附加目标是开发新的，灵活的方法来研究这些问题，结合极值集理论，高维几何和概率的工具来研究图和超图的结构。在最近的工作中，我对这些工具的相互作用进行了新颖的利用，解决了拉姆齐理论和极值超图理论中的几个著名问题，但这些想法还有更大的潜力。下文提出的研究将大大加强这些联系，实现上述目标，并在这一重要领域提供新的认识。

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.