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Abstract for Fr\'ed\'eric Maire, On the shadow of squashed families of $k$-sets

The {\it shadow} of a collection   ${\cal A}$ of $k$-sets 
is defined as the collection
of the $(k-1)$-sets which are contained in at least one $k$-set of 
 ${\cal A}$. Given $|{\cal A}|$, the size of the shadow is minimum 
when ${\cal A}$ is the family of the first $k$-sets in 
{\it 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.

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