  In this paper we study variations of an old result by M\"{u}ller, Reiterman,
and the last author stating that a countable graph has a subgraph with infinite
degrees if and only if in any labeling of the vertices (or edges) of this graph
by positive integers we can always find an infinite increasing path. We study
corresponding questions for hypergraphs and directed graphs. For example we
show that the condition that a hypergraph contains a subhypergraph with
infinite degrees is equivalent to the condition that any vertex labeling
contains an infinite increasing loose path. We also find an equivalent
condition for a graph to have a property that any vertex labeling with positive
integers contains a path of arbitrary finite length, and we study related
problems for oriented graphs and labelings with $\mathbb{Z}$ (instead of
$\mathbb{N}$). For example, we show that for every simple hypergraph, there is
a labelling of its edges by $\mathbb{Z}$ that forbids one-way infinite
increasing paths.
