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

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arxiv 2104.00045 v1 pith:XD2FPHED submitted 2021-03-31 math.CO

classification math.CO
keywords configurationsoperationsexistsgeometricproducethereunbaumanalysis
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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 $k$.

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