伪半平面的离散Helly型定理
Discrete Helly-type theorems for pseudohalfplanes
摘要：
本文证明了伪半平面的离散Helly型定理，推广了Jensen，Joshi和Ray关于半平面的最新结果.其中，我们表明，给定一个家庭的pseudohavelanes $\cal H$和一组点$P$，如果每三个pseudohavelanes有一个共同的点在$P$然后存在一组至多两个点，击中每一个pseudohavelanes的$\cal H$。我们还证明了，如果每三个点的$P$包含在一个伪半平面的$\cal H$，那么有两个伪半平面的$\cal H$，覆盖所有点的$P$。为了证明我们的结果，我们把伪半平面超图，定义其极值顶点，并表明，这些行为在许多方面作为一组点的凸包的边界上的点。我们的方法是纯组合的。此外，我们确定了所考虑的超图族的最大可能色数。
英文摘要：
We prove discrete Helly-type theorems for pseudohalfplanes, which extend recent results of Jensen, Joshi and Ray about halfplanes. Among others we show that given a family of pseudohalfplanes $\cal H$ and a set of points $P$, if every triple of pseudohalfplanes has a common point in $P$ then there exists a set of at most two points that hits every pseudohalfplane of $\cal H$. We also prove that if every triple of points of $P$ is contained in a pseudohalfplane of $\cal H$ then there are two pseudohalfplanes of $\cal H$ that cover all points of $P$. To prove our results we regard pseudohalfplane hypergraphs, define their extremal vertices and show that these behave in many ways as points on the boundary of the convex hull of a set of points. Our methods are purely combinatorial. In addition we determine the maximal possible chromatic number of the regarded hypergraph families.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
