论文标题
公制超图和度量线等价
Metric hypergraphs and metric-line equivalences
论文作者
论文摘要
在公制空间$ m =（x，d）$中，我们说$ v $在$ u $和$ w $之间，如果$ d（u，w）= d（u，v）+d（v）+d（v，w）$。服用所有三倍$ \ {u，v，w \} $，以便$ v $在$ u $和$ w $之间，可以将一个3-均匀的超图与每个有限的度量公制$ m $相关联。解决有关有限度量空间的一些基本开放问题的努力促使一项努力更好地了解这些相关的超图。为了回答Arxiv中提出的问题：1112.0376，我们提出了一个无限的超图家族，这些家族是非现有的，即它们并非来自任何度量空间。 与度量空间相关的另一个基本结构是顶点集合的二进制等效性，如果它们诱导同一条线，则两个对在同一类中。来自某些度量空间的等效性是度量线等效性。我们提出了一个所谓的障碍的无限家族，即二进制等价，可以防止等价成为指标。
In a metric space $M=(X,d)$, we say that $v$ is between $u$ and $w$ if $d(u,w)=d(u,v)+d(v,w)$. Taking all triples $\{u,v,w\}$ such that $v$ is between $u$ and $w$, one can associate a 3-uniform hypergraph with each finite metric space $M$. An effort to solve some basic open questions regarding finite metric spaces has motivated an endeavor to better understand these associated hypergraphs. In answer to a question posed in arXiv:1112.0376, we present an infinite family of hypergraphs that are non-metric, i.e., they don't arise from any metric space. Another basic structure associated with a metric space is a binary equivalence on the vertex set, where two pairs are in the same class if they induce the same line. An equivalence that comes from some metric space is a metric-line equivalence. We present an infinite family of so called obstacles, that is, binary equivalences that prevent an equivalence from being a metric-line equivalence.
