Chvátal's conjecture: a proof from The Book

Chvátal conjectured that every downset has a maximum-size intersecting family which is a star, that is, consists of all members of the family containing a fixed element. Recently, Chang, Liu and Liu gave a proof of this conjecture, which follows as a corollary of more general results such as Kleitman’s conjecture and a version of Kahn’s conjecture (which they also prove). We give a short, direct (spectral) proof of Chvátal’s conjecture, and also prove a strengthening concerning the projection packing number. We also propose two spectral Chvátal conjectures which are motivated by extensive numerical experiments.