  For $n > 2k \geq 4$ we consider intersecting families $\mathcal F$ consisting
of $k$-subsets of $\{1, 2, \ldots, n\}$. Let $\mathcal I(\mathcal F)$ denote
the family of all distinct intersections $F \cap F'$, $F \neq F'$ and $F, F'\in
\mathcal F$. Let $\mathcal A$ consist of the $k$-sets $A$ satisfying $|A \cap
\{1, 2, 3\}| \geq 2$. We prove that for $n \geq 50 k^2$ $|\mathcal I(\mathcal
F)|$ is maximized by $\mathcal A$.
