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 polytope, 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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We are indebted to the anonymous reviewer for several insightful remarks.
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David Conlon: Research supported by NSF Award DMS-2054452. Jeck Lim: Research partially supported by the NUS Overseas Graduate Scholarship
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Conlon, D., Lim, J. Fixing a Hole. Discrete Comput Geom 70, 1551–1570 (2023). https://doi.org/10.1007/s00454-023-00561-6
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DOI: https://doi.org/10.1007/s00454-023-00561-6