Extremal results on intersection graphs of boxes in $R^d$

The main purpose of this paper is to study extremal results on the intersection graphs of boxes in ${\mathbb{R}}^d$. We calculate exactly the maximal number of intersecting pairs in a family $\text{F}$ of $n$ boxes in ${\mathbb{R}}^d$ with the property that no $k+1$ boxes in $\text{F}$ 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.