
Problem Let
be a graph,
a countable end of
, and
an infinite set of pairwise disjoint
-rays in
. Prove that there is a set
of pairwise disjoint
-rays that devours
such that the set of starting vertices of rays in
equals the set of starting vertices of rays in
.











We say that a set of rays devours the end
if every ray in
meets some ray in
. An end is countable if there is a countable set of rays devouring it.
If is a finite set of rays then it is not hard to prove (see [G]) that this problem has a positive answer:
Theorem For every graph
and every countable end
of
, if
has a set
of
pairwise disjoint
-rays, then it also has a set
of
pairwise disjoint
-rays that devours
. Moreover,
can be chosen so that its rays have the same starting vertices as the rays in~
.













Bibliography
*[G] A. Georgakopoulos, Infinite Hamilton Cycles in Squares of Locally Finite Graphs, Preprint.
* indicates original appearance(s) of problem.