The previous talk in the series is here. The next talk in the series is here.
Let X be a subset of ℝn such that there are only two possible distances between distinct elements of X. How large can X be? An example of such a set is the set of all 01 vectors with precisely two 1s, which has size n(n-1)/2. In this video I show how to prove an upper bound of (n+1)(n+4)/2 by associating a polynomial with each element of X, showing that those polynomials are linearly independent, and showing that they live in a space of dimension (n+1)(n+4)/2. This will be the first of several dimension arguments that I shall present in the course.
This video was produced by Tim Gowers as part of his Part III course at the University of Cambridge. Printed notes for this course are available here.