  The main purpose of this paper is to study extremal results on the
intersection graphs of boxes in $\R^d$. We calculate exactly the maximal number
of intersecting pairs in a family $\F$ of $n$ boxes in $\R^d$ with the property
that no $k+1$ boxes in $\F$ have a point in common. This allows us to improve
the known bounds for the fractional Helly theorem for boxes. We also use the
Fox-Gromov-Lafforgue-Naor-Pach results to derive a fractional Erd\H{o}s-Stone
theorem for semi-algebraic graphs in order to obtain a second proof of the
fractional Helly theorem for boxes.
