Inventiones mathematicae

, Volume 176, Issue 1, pp 1-62

First online:

Menger’s theorem for infinite graphs

  • Ron AharoniAffiliated withDepartment of Mathematics, Technion Email author 
  • , Eli BergerAffiliated withDepartment of Mathematics, Faculty of Science and Science Education, Haifa University

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We prove that Menger’s theorem is valid for infinite graphs, in the following strong version: let A and B be two sets of vertices in a possibly infinite digraph. Then there exist a set \(\mathcal{P}\) of disjoint AB paths, and a set S of vertices separating A from B, such that S consists of a choice of precisely one vertex from each path in \(\mathcal{P}\). This settles an old conjecture of Erdős.