Connected geometric (n_k) configurations exist for almost all n

In a series of papers and in his 2009 book on configurations Branko Grunbaum described a sequence of operations to produce new (n4) configurations from various input configurations. These operations were later called the “Grunbaum Incidence Calculus”. We generalize two of these operations to produce operations on arbitrary (nk) configurations. Using them, we show that for any k there exists an integer Nk such that for any n ≥ Nk there exists a geometric (nk) 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.