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.
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