TRM intern Rebekah Glaze explains Van der Waerden’s Theorem on the existence of Arithmetic Progressions in sets, using the example of strings of coloured beads.

We begin by looking at the specific case of 2 colours, where we are trying to guarantee a pattern of 3 beads of the same colour which are equally spaced. This is then generalised to n colours, and a pattern of length k, which is the statement of Van der Waerden’s Theorem. The case of 3 colours (n=3), and arithmetic progressions of length 3 (k=3), is then shown to be true via a proof by construction. The length of chain used in the proof is much larger than the minimum required, but as the theorem only requires the existence of a sufficiently long chain, our proof suffices. The question of finding the minimum length of chain for specific cases of n colours, and arithmetic progression length k, remains an unsolved problem apart from in a handful of cases.