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arXiv · 2306.00414

On $k$-neighborly reorientations of oriented matroids

Rangel Hernández-Ortiz·Kolja Knauer·Luis Pedro Montejano
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Abstract

We study the existence and the number of $k$-neighborly reorientations of an oriented matroid. This leads to $k$-variants of McMullen's problem and Roudneff's conjecture, the case $k=1$ being the original statements on complete cells in arrangements. Adding to results of Larman and Garc\'ia-Col\'in, we provide new bounds on the $k$-McMullen's problem and prove the conjecture for several ranks and $k$ by computer. Further, we show that $k$-Roudneff's conjecture for fixed rank and $k$ reduces to a finite case analyse. As a consequence we prove the conjecture for odd rank $r$ and $k=\frac{r-1}{2}$ as well as for rank $6$ and $k=2$ with the aid of the computer.

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Rangel Hernández-Ortiz, Kolja Knauer, Luis Pedro Montejano. 2023-06-01. On $k$-neighborly reorientations of oriented matroids. https://arxiv.org/abs/2306.00414

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