Tight bounds for Katona's shadow intersection theorem

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 $n$.