  Let $A$ and $B$ be finite sets and consider a partition of the \emph{discrete
box} $A \times B$ into \emph{sub-boxes} of the form $A' \times B'$ where $A'
\subset A$ and $B' \subset B$. We say that such a partition has the
$(k,\ell)$-piercing property for positive integers $k$ and $\ell$ if every
\emph{line} of the form $\{a\} \times B$ intersects at least $k$ sub-boxes and
every line of the form $A \times \{b\}$ intersects at least $\ell$ sub-boxes.
We show that a partition of $A \times B$ that has the $(k, \ell)$-piercing
property must consist of at least $(k-1)+(\ell-1)+\left\lceil
2\sqrt{(k-1)(\ell-1)} \right\rceil$ sub-boxes. This bound is nearly sharp (up
to one additive unit) for every $k$ and $\ell$. As a corollary we get that the
same bound holds for the minimum number of vertices of a graph whose edges can
be colored red and blue such that every vertex is part of red $k$-clique and a
blue $\ell$-clique.
