Van der Waerden's Theorem - Finding Patterns in Sets

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. The content in the video was inspired by a lecture from Prof. Imre Leader at the University of Cambridge. Produced by TRM intern Rebekah Glaze with assistance from Dr Tom Crawford. Rebekah is a second year mathematics undergraduate at the University of Oxford. Tom is Public Engagement Lead at the Oxford University Department of Continuing Education: https://www.conted.ox.ac.uk/profiles/tom-crawford If you would like to take part in the Tom Rocks Maths intern scheme, please get in touch using the contact form on the TRM website: https://tomrocksmaths.com/contact/ You can find more videos made by interns working with Tom on the designated playlist here: https://www.youtube.com/playlist?list=PLMCRxGutHqfn8YkbswsI3HDOaVQDUTZMp For more maths content check out Tom's website https://tomrocksmaths.com/ You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. https://www.facebook.com/tomrocksmaths/ https://twitter.com/tomrocksmaths https://www.instagram.com/tomrocksmaths/ Get your Tom Rocks Maths merchandise here: https://beautifulequations.net/collections/tom-rocks-maths