# A topological proof of Van der Waerden's theorem

## Matthew Gwynne

(Swansea)

(Joint event with the MRes seminar.)

Van der Waerden's theorem states that for any finite partitioning of the natural numbers, there must be one part which contains arithmetic progressions of arbitrary length. This theorem provides an example of a particular kind of Ramsey-type problem, and provides a basis for discussion of the finite version of the theorem, leading to the notion of the Van der Waerden numbers.

In this talk, a proof is given of Van der Waerden's theorem using topological notions, and algebraic structures over "ultrafilters". Such ultrafilters, in this case, are used to generalise the notion of addition and arithmetic sequences within the natural numbers and motivate the proof.

The plan is to give the talk in two parts:

2-2:45 General, for a wider audience.

3-3:45 Technical details, for those who are specifically interested

**Thursday 19th March 2009, 14:00**

Robert Recorde Room

Department of Computer Science