Australas. J. Combin. 52 (2012), 205–207. PDF

How large can an independent set in the discrete cube be if it contains equal numbers of sets of even and odd size? Take odd sets starting from the bottom of the cube, and even sets starting from the top. Proving that this works uses an isoperimetric inequality: if you know the proof of Harper’s theorem that uses codimension 1 compressions then you know how to prove the inequality that’s quoted without proof in this paper.