Imre Leader: Partition Regular Equations

A finite or infinite matrix M is called ‘partition regular’ if whenever the natural numbers are finitely coloured there exists a monochromatic vector x with Mx = 0. Many of the classical results of Ramsey theory, such as van der Waerden's theorem or Schur’s theorem, may be naturally rephrased as assertions that certain matrices are partition regular.

While the structure of finite partition regular matrices is well understood, little is known in the infinite case. In this talk we will review some known results and then proceed to some recent developments.

The talk will not assume any previous knowledge of the area.