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arXiv · 1410.0467

A fractional Helly theorem for boxes

I. Bárány·F. Fodor·A. Martínez-Pérez·L. Montejano·D. Oliveros·A. Pór
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Abstract

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

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I. Bárány, F. Fodor, A. Martínez-Pérez, L. Montejano, D. Oliveros, A. Pór. 2014-10-02. A fractional Helly theorem for boxes. https://doi.org/10.1016/j.comgeo.2014.09.007

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