Topics in Combinatorics lecture 16.9 --- Sets without orthogonal pairs, and Borsuk's conjecture

How large can a subset of the unit sphere of R^n 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).

0:00 A bound on the measure of a subset of the unit sphere in R^n that contains no pair of orthogonal vectors
18:46 Introductory discussion of Borsuk's conjecture
23:05 Kahn and Kalai's amazing solution to the conjecture

Notes for the course available here:

A question sheet (some of the questions quite hard) on the last third of the course is here:

[strawandherb]

It feels weird to finish such a course without the very much deserved clapping.

So here is an electronic substitute...

Thank you.
Thank you very much!