  Extending a classical theorem of Sperner, we characterize the integers $m$
such that there exists a maximal antichain of size $m$ in the Boolean lattice
$B_n$, that is, the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion.
As an important ingredient in the proof, we initiate the study of an extension
of the Kruskal-Katona theorem which is of independent interest. For given
positive integers $t$ and $k$, we ask which integers $s$ have the property that
there exists a family $\mathcal F$ of $k$-sets with $\lvert\mathcal F\rvert=t$
such that the shadow of $\mathcal F$ has size $s$, where the shadow of
$\mathcal F$ is the collection of $(k-1)$-sets that are contained in at least
one member of $\mathcal F$. We provide a complete answer for $t\leqslant k+1$.
Moreover, we prove that the largest integer which is not the shadow size of any
family of $k$-sets is $\sqrt 2k^{3/2}+\sqrt[4]{8}k^{5/4}+O(k)$.
