New bounds on the existence of \((n_{5})\) and \((n_{6})\) configurations: the Gr\"{u}nbaum Calculus revisited

The "Gr\"unbaum Incidence Calculus" is the common name of a collection of operations introduced by Branko Gr\"unbaum to produce new \((n_{4})\) configurations from various input configurations. In a previous paper, we generalized two of these operations to produce operations on a...

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Bibliographic Details
Published in:arXiv.org 2022-05
Main Authors: Leah Wrenn Berman, Gévay, Gábor, Pisanski, Tomaž
Format: Article
Language:English
Subjects:
Calculus
Configurations
Online Access:Get full text
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Summary:The "Gr\"unbaum Incidence Calculus" is the common name of a collection of operations introduced by Branko Gr\"unbaum to produce new \((n_{4})\) configurations from various input configurations. In a previous paper, we generalized two of these operations to produce operations on arbitrary \((n_k)\) configurations, and we showed that for each \(k\), there exists an integer \(N_{k}\) such that for all \(n \geq N_{k}\), there exists at least one \((n_{k})\) configuration, with current records \(N_{5}\leq 576\) and \(N_{6}\leq 7350\). In this paper, we further extend the Gr\"unbaum calculus; using these operations, as well as a collection of previously known and novel ad hoc constructions, we refine the bounds for \(k = 5\) and \(k = 6\). Namely, we show that \(N_5 \leq 166\) and \(N_{6}\leq 585\).
ISSN:2331-8422