  Let $n\geqslant 4$ be a natural number, and let $K$ be a set $K\subseteq
[n]:={1,2,...,n}$. We study the problem to find the smallest possible size of a
maximal family $\mathcal{A}$ of subsets of $[n]$ such that $\mathcal{A}$
contains only sets whose size is in $K$, and $A\not\subseteq B$ for all
${A,B}\subseteq\mathcal{A}$, i.e. $\mathcal{A}$ is an antichain. We present a
general construction of such antichains for sets $K$ containing 2, but not 1.
If $3\in K$ our construction asymptotically yields the smallest possible size
of such a family, up to an $o(n^2)$ error. We conjecture our construction to be
asymptotically optimal also for $3\not\in K$, and we prove a weaker bound for
the case $K={2,4}$. Our asymptotic results are straightforward applications of
the graph removal lemma to an equivalent reformulation of the problem in
extremal graph theory which is interesting in its own right.
