The previous talk in the series is here.
How large can a subset of the unit sphere of ℝn be if it contains no pair of orthogonal vectors? The Frankl-Wilson theorem (in the version proved in the previous video) implies that it must be exponentially small. Another extraordinary application of the Frankl-Wilson theorem is a result of Jeff Kahn and Gil Kalai, which solved a 60-year-old conjecture of Borsuk. Both results are presented in this video, which is the final one of the course (except that I may add some concluding remarks at some point). We start with a bound on the measure of a subset of the unit sphere in ℝn that contains no pair of orthogonal vectors. Then there is an introductory discussion of Borsuk’s conjecture, and finally Kahn and Kalai’s amazing solution to the conjecture.
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.