In many different areas of mathematics (such as number theory, discrete geometry and combinatorics), one is often presented with a large "unstructured" object, and asked to find a smaller "structured" object inside it. One of the earliest and most influential examples of this phenomenon was the theorem of Ramsey, proved in 1930, which states that if n = n(k) is large enough, then in any red-blue colouring of the edges of the complete graph on n vertices, there exists a monochromatic clique on k vertices. In this talk I will discuss some of the questions, ideas, and new techniques that were inspired by this theorem, and mention some recent progress on one of the central problems in the area: bounding the so-called "diagonal" Ramsey numbers.
Based on joint work with Marcelo Campos, Simon Griffiths and Julian Sahasrabudhe.
A wine reception in the Central Core will follow the lecture
