The Erdős-Menger theorem is the infinite form of Menger's n-arc theorem. For disjoint sets 
and 
of vertices in a possibly
infinite directed graph, there are a family 
of pairwise vertex-disjoint finite paths from 
to 
and a separating set 
consisting of exactly one graph
vertex from each graph path in 
(Aharoni and Berger 2009). Every graph
path from 
to 
meets 
.
Aharoni and Berger (2009) proved this assertion, formerly called the Erdős-Menger conjecture. The condition that 
selects a graph vertex on
each chosen graph path is essential in the infinite
setting; equality of the two cardinalities alone
would be weaker.
This is problem 599 in the Erdős problems collection (Bloom 2026).
