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

TL;DR

This paper refines bounds on the existence of certain geometric configurations using an extended calculus, showing that for 5- and 6-configurations, such configurations exist for all sufficiently large n, with improved bounds.

Contribution

The authors extend Gr"unbaum's calculus to improve bounds on the minimal n for which (n_5) and (n_6) configurations exist, reducing previous bounds significantly.

Findings

N_5  166

N_6  585

Established existence for all n  N_k for k=5,6 beyond these bounds

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 $N_{6}\leq 585$.