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arXiv · 1412.8190

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

A. Martínez-Pérez·L. Montejano·D. Oliveros
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Abstract

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 Erdős-Stone theorem for semi-algebraic graphs in order to obtain a second proof of the fractional Helly theorem for boxes.

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A. Martínez-Pérez, L. Montejano, D. Oliveros. 2015-01-18. Extremal results on intersection graphs of boxes in $R^d$. https://arxiv.org/abs/1412.8190

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