Yuval Wigderson: Bounds on Ramsey numbers

Yuval Wigderson (ETH)

Ramsey's theorem roughly states that "complete disorder is impossible": any sufficiently large system, no matter how complex and unstructured, must contain large, structured pieces. Since its proof almost 100 years ago, Ramsey's theorem and related statements have become fundamental results in combinatorics, functional analysis, geometry, number theory, theoretical computer science, and many other fields of mathematics.

A natural question arising from Ramsey's theorem is "how large is sufficiently large?", which leads to the notion of Ramsey numbers. The problem of finding good upper and lower bounds on Ramsey numbers has proved to be surprisingly difficult, but has seen some remarkable breakthroughs in recent years. In this talk, I will discuss the background and motivation for this problem, and highlight some of the new ideas that have been introduced in its study.