标题： 关于有向拟阵的$k$-邻接重新定向 显示英文标题

标题： On $k$-neighborly reorientations of oriented matroids

摘要： 我们研究有向拟阵的$k$-邻域重定向的存在性和数量。 这导致了$k$的 McMullen 问题和 Roudneff 猜想的变体，$k=1$的情况是关于排列中完整单元的原始陈述。 在 Larman 和 García-Colín 的结果基础上，我们提供了$k$-McMullen 问题的新界限，并通过计算机证明了几个秩和$k$的猜想。 此外，我们表明对于固定秩和$k$，$k$-Roudneff 猜想可归结为有限情况分析。 作为结果，我们借助计算机证明了奇数阶$r$和$k=\frac{r-1}{2}$的猜想，以及阶$6$和$k=2$的猜想。 显示英文摘要

摘要： 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.