Katona's intersection theorem: Four proofs

It is known from a previous paper [3] that Katona's Intersection Theorem follows from the Complete Intersection Theorem by Ahlswede and Khachatrian via a Comparison Lemma. It also has been proved directly in [3] by the pushing-pulling method of that paper. Here we add a third proof via a new (k, k+1)-shifting technique, whose impact will be exploared elsewhere. The fourth and last of our proofs is a gift from heaven for Gyula's birthday.