Tight bounds for Katona’s shadow intersection theorem

Abstract

A fundamental result in extremal set theory is Katona’s shadow intersection theorem, which extends the Kruskal–Katona theorem by giving a lower bound on the size of the shadow of an intersecting family of k-sets in terms of its size. We improve this classical result and a related result of Ahlswede, Aydinian and Khachatrian by proving tight bounds for families that can be quite small. For example, when k=3 our result is sharp for all families with n points and at least 3n−7 triples.Katona’s theorem was extended by Frankl to families with matching number s. We improve Frankl’s result by giving tight bounds for large n.