An old result of M\"uller and R\"odl states that a countable graph $G$ has a subgraph whose vertices all have infinite degree if and only if for any vertex labeling of $G$ by positive integers, an infinite increasing path can be found. They asked whether an analogous equivalence holds for edge labelings, which Reiterman answered in the affirmative. Recently, Arman, Elliott, and R\"odl extended this problem to linear $k$-uniform hypergraphs $H$ and generalized the original equivalence for vertex labelings. They asked whether Reiterman's result for edge labelings can similarly be extended. We confirm this for the case where $H$ admits only finitely many Berge cycles.
翻译:M\“Uller”和R\“Odl”的旧结果指出,一个可计数的图表$G$有一个子图,它的脊椎都具有无限程度,如果而且只有在任何以正数整数标注$G$的顶点上,才能找到一条无限增加的路径。他们询问边缘标签是否具有类似的等值,而Reytermanan的回答是肯定的。最近,Arman, Elliott 和R\“odl ” 将这一问题扩大到直线$k$-单体高射法$H$,并普及了顶端标签的原始等值。他们询问Reiteman的边缘标签结果能否同样得到延长。我们确认,对于美元只允许许多Berge周期的病例,只要$就只有有限的H值。