In any sufficiently large party, you are guaranteed to find either three mutual friends or three mutual strangers—and mathematicians want to know the same kind of unavoidable patterns in numbers and groups. This project tackles a gap in Ramsey theory: while researchers know that certain patterns must appear in large enough mathematical structures, they lack general methods to prove this for many different settings. The PI aims to develop broad, reusable approaches, starting with arithmetic progressions hidden inside the prime numbers, then extending to whole numbers and abelian groups. The core idea is to translate these number-theory problems into problems about hypergraphs—mathematical networks of connections—and then solve them using combinatorial tools. If successful, the project will give mathematicians a powerful new toolkit for proving pattern-existence results across algebra and number theory. This is fundamental science with no immediate practical application. But Ramsey theory has historically fed into theoretical computer science, data compression, and network design. A deeper understanding of unavoidable patterns in mathematical structures could, over the long term, inform algorithms that detect structure in large datasets or design more efficient communication networks.
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The prototypical question in Ramsey theory asks how large a mathematical structure must be in order to guarantee a particular type of pattern emerges. For example, a special case of Ramsey theory for graphs can be rephrased as follows: in any party of six people there is always a set of three people who know each other or a set of three people who do not know each other; note, this is not true in general for a party of five people. Ramsey-type results occur in a wide range of settings, including in Graph Theory, Number Theory and Theoretical Computer Science. In recent decades, there has been significant interest in the typical Ramsey properties mathematical structures of a given density possess. Whilst there have been several recent breakthroughs in this area for specific mathematical structures, there is still a need to develop general approaches that are applicable to a wide variety of settings. The aim of this project is to capitalise on the personal momentum of the PI to develop such general approaches in arithmetic settings. The initial phase of the project will concern arithmetic progressions in the prime numbers, before moving on to consider Ramsey properties of collections of whole numbers more generally. Progress on these topics will then be used as a spring-board for studying analogous questions in the wider setting of abelian groups. The project will study the combinatorial essence of these problems from Algebra and Number Theory. Indeed, these questions will be attacked by transforming them into problems concerning mathematical networks, specifically, hypergraphs. Our methodology will demonstrate that, as long as so-called supersaturation occurs, one can strip back the arithmetic structure in sets of integers or abelian groups, and instead simply consider the corresponding hypergraph problem. Thus, a long-term aim of the project is to provide researchers in these areas with the combinatorial tools needed to attack such questions.
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