The previous talk in the series is here. The next talk in the series is here.
How many subsets of {1,…,n} can you choose if no set in your collection is allowed to be a subset of another? An obvious construction is to take all sets of size n/2 (or (n-1)/2 if n is odd). Sperner’s theorem tells us that this is the best we can do. It has a miraculously short and simple proof by double counting. First, we have terminology concerning the discrete cube, then give Sperner’s theorem, and finally consider the equality case.
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.