New bounds on the existence of $(n_5)$ and $(n_6)$ configurations: the Grünbaum Calculus revisited

Abstract

The "Grünbaum Incidence Calculus" is the common name of a collection of operations introduced by Branko Grünbaum 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 $n\ge N_k$, there exists at least one $(n_k)$ configuration, with current records $N_5\le 576$ and $N_6\le 7350$. In this paper, we further extend the Grünbaum 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 $N_5\le 166$ and $N_6\le 585$.