  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 $n$.
