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arXiv · 2108.07087

Fixing a hole

David Conlon·Jeck Lim
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Abstract

We show that any finite $S \subset \mathbb{R}^d$ in general position has arbitrarily large supersets $T \supseteq S$ in general position with the property that $T$ contains no empty convex polygon, or hole, with $C_d$ points, where $C_d$ is an integer that depends only on the dimension $d$. This generalises results of Horton and Valtr which treat the case $S = \emptyset$. The key step in our proof, which may be of independent interest, is to show that there are arbitrarily small perturbations of the set of lattice points $[n]^d$ with no large holes.

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David Conlon, Jeck Lim. 2022-11-10. Fixing a hole. https://arxiv.org/abs/2108.07087

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