  The product dimension of a graph $G$ is the minimum possible number of proper
vertex colorings of $G$ so that for every pair $u,v$ of non-adjacent vertices
there is at least one coloring in which $u$ and $v$ have the same color. What
is the product dimension $Q(s,r)$ of the vertex disjoint union of $r$ cliques,
each of size $s$? Lov\'asz, Ne\v{s}et\v{r}il and Pultr proved in 1980 that for
$s=2$ it is $(1+o(1)) \log_2 r$ and raised the problem of estimating this
function for larger values of $s$. We show that for every fixed $s$, the answer
is still $(1+o(1)) \log_2 r$ where the $o(1)$ term tends to $0$ as $r$ tends to
infinity, but the problem of determining the asymptotic behavior of $Q(s,r)$
when $s$ and $r$ grow together remains open. The proof combines linear
algebraic tools with the method of Gargano, K\"orner, and Vaccaro on Sperner
capacities of directed graphs.
