On the shadow of squashed families of \(k\)-sets (Q1897683)

Summary: The shadow of a collection \(\mathcal A\) of \(k\)-sets is defined as the collection of the \((k- 1)\)-sets which are contained in at least one \(k\)- set of \(\mathcal A\). Given \(| {\mathcal A}|\), the size of the shadow is minimum when \(\mathcal A\) is the family of the first \(k\)-sets in squashed order (by definition, a \(k\)-set \(A\) is smaller than a \(k\)-set \(B\) in the squashed order if the largest element of the symmetric difference of \(A\) and \(B\) is in \(B\)). We give a tight upper bound and an asymptotic formula for the size of the shadow of squashed families of \(k\)-sets.