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arXiv · 2204.11986

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

Leah Wrenn Berman·Gábor Gévay·Tomaž Pisanski
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Abstract

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 $N_{6}\leq 585$.

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Leah Wrenn Berman, Gábor Gévay, Tomaž Pisanski. 2022-04-25. New bounds on the existence of $(n_{5})$ and $(n_{6})$ configurations: the Gr\"{u}nbaum Calculus revisited. https://arxiv.org/abs/2204.11986

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