English

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

Combinatorics 2021-04-02 v1

Abstract

In a series of papers and in his 2009 book on configurations Branko Gr\"unbaum described a sequence of operations to produce new (n4)(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 (nk)(n_{k}) configurations. Using them, we show that for any kk there exists an integer NkN_k such that for any nNkn \geq N_k there exists a geometric (nk)(n_k) configuration. We use empirical results for k=2,3,4k = 2, 3, 4, and some more detailed analysis to improve the upper bound for larger values of kk.

Keywords

Cite

@article{arxiv.2104.00045,
  title  = {Eventually, geometric $(n_{k})$ configurations exist for all $n$},
  author = {Leah Wrenn Berman and Gábor Gévay and Tomaž Pisanski},
  journal= {arXiv preprint arXiv:2104.00045},
  year   = {2021}
}
R2 v1 2026-06-24T00:44:56.198Z