Eventually, geometric $(n_k)$ configurations exist for all $n$

Abstract

In a series of papers and in his 2009 book on configurations Branko Grünbaum described a sequence of operations to produce new $(n_4)$ configurations from various input configurations. These operations were later called the "Grünbaum 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 $n\ge N_k$ 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 $k$.