  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.
