  Let $\mathcal{A}$ and $\matchcal{B}$ denote two families of subsets of an
$n$-element set. The pair $(\mathcal{A},\mathcal{B})$ is said to be
$\ell$-cross-intersecting iff $|A\cap B| = \ell$ for all $A\in\mathcal{A}$ and
$B\in\mathcal{B}$. Denote by $P_\ell(n)$ the maximum value of
$|\mathcal{A}||\mathcal{B}|$ over all such pairs. The best known upper bound on
$P_\ell(n)$ is $\Theta(2^n)$, by Frankl and R\"{o}dl. For a lower bound,
Ahlswede, Cai and Zhang showed, for all $n \geq 2\ell$, a simple construction
of an $\ell$-cross-intersecting pair $(\mathcal{A},\mathcal{B})$ with
$|\mathcal{A}||\mathcal{B}| =
\binom{2\ell}{\ell}2^{n-2\ell}=\Theta(2^n/\sqrt{\ell})$, and conjectured that
this is best possible. Consequently, Sgall asked whether or not $P_\ell(n)$
decreases with $\ell$.
  In this paper, we confirm the above conjecture of Ahlswede et al. for any
sufficiently large $\ell$, implying a positive answer to the above question of
Sgall as well. By analyzing the linear spaces of the characteristic vectors of
$\mathcal{A},\mathcal{B}$ over $\mathbb{R}$, we show that there exists some
$\ell_0>0$, such that $P_\ell(n) \leq \binom{2\ell}{\ell}2^{n-2\ell}$ for all
$\ell \geq \ell_0$. Furthermore, we determine the precise structure of all the
pairs of families which attain this maximum.
