  We consider the following list coloring with separation problem: Given a
graph $G$ and integers $a,b$, find the largest integer $c$ such that for any
list assignment $L$ of $G$ with $|L(v)|= a$ for any vertex $v$ and $|L(u)\cap
L(v)|\le c$ for any edge $uv$ of $G$, there exists an assignment $\varphi$ of
sets of integers to the vertices of $G$ such that $\varphi(u)\subset L(u)$ and
$|\varphi(v)|=b$ for any vertex $u$ and $\varphi(u)\cap \varphi(v)=\emptyset$
for any edge $uv$. Such a value of $c$ is called the separation number of
$(G,a,b)$. Using a special partition of a set of lists for which we obtain an
improved version of Poincar\'e's crible, we determine the separation number of
the complete graph $K_n$ for some values of $a,b$ and $n$, and prove bounds for
the remaining values.
