Problems in Ramsey theory

A red-blue colouring of a complete graph on N vertices must always contain a monochromatic copy of a given smaller graph H, provided N is large enough—and mathematicians want to know the smallest, most efficient graphs that guarantee this property. This project tackles two open problems in Ramsey theory, a branch of combinatorics that studies unavoidable patterns in large structures. The first asks how sparse a graph can be—specifically, how low its minimum degree can fall—while still forcing a monochromatic H in every red-blue edge colouring. The second asks for the maximum number of edge-disjoint monochromatic copies of H that can appear in a coloured complete graph; the researcher has already solved this for triangles and now seeks to extend the result to other graphs. This is fundamental, curiosity-driven mathematics. It has no immediate practical application. But Ramsey theory underpins parts of computer science, including network design and algorithm analysis, where understanding unavoidable patterns in large data sets can inform error-correcting codes or distributed computing. Past work in this area has also fed into the mathematics of social networks and scheduling problems. The project will train a PhD student in rigorous combinatorial reasoning, contributing to the UK’s research capacity in pure mathematics.