Eventually, geometric $(n_{k})$ configurations exist for all $n$

TL;DR

This paper proves that geometric $(n_k)$ configurations exist for all sufficiently large $n$, generalizing previous operations and providing bounds for their existence across all $k$ values.

Contribution

It generalizes the Grünbaum Incidence Calculus to arbitrary $(n_k)$ configurations and establishes existence results for all large $n$ for any fixed $k$.

Findings

Existence of geometric $(n_k)$ configurations for all sufficiently large $n$.

Generalization of operations to produce new configurations for any $(n_k)$.

Improved bounds for the minimal $n$ for configurations with larger $k$.

Abstract

In a series of papers and in his 2009 book on configurations Branko Gr\"unbaum described a sequence of operations to produce new $(n_{4})$ configurations from various input configurations. These operations were later called the "Gr\"unbaum Incidence Calculus". We generalize two of these operations to produce operations on arbitrary $(n_{k})$ configurations. Using them, we show that for any $k$ there exists an integer $N_k$ such that for any $n \geq N_k$ there exists a geometric $(n_k)$ configuration. We use empirical results for $k = 2, 3, 4$, and some more detailed analysis to improve the upper bound for larger values of $k$.