A fractional Helly theorem for boxes

Let $\mathcal{F}$ be a family of $n$ axis-parallel boxes in $\mathbb{R}^d$ and $α\in (1-1/d,1]$ a real number. There exists a real number $β(α)>0$ such that if there are $α{n\choose 2}$ intersecting pairs in $\mathcal{F}$, then $\mathcal{F}$ contains an intersecting subfamily of size $βn$. A simple example shows that the above statement is best possible in the sense that if $α\leq 1-1/d$, then there may be no point in $\mathbb{R}^d$ that belongs to more than $d$ elements of $\mathcal{F}$.

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