The board / graph theory / #599
Erdős Problem #599
Erdős-Menger conjecture
PROVEDdatabase updated 2025-08-31
The statement as posed is true, and a proof exists.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
**Erdős Problem 599** (the Erdős–Menger conjecture). Let \(G\) be a (possibly infinite) graph and let \(A, B\) be disjoint independent sets of vertices. Must there exist a family \(P\) of pairwise vertex-disjoint paths from \(A\) to \(B\), and a set \(S\) of vertices containing exactly one vertex from each path in \(P\), such that every path from \(A\) to \(B\) contains at least one vertex of \(S\)? For finite \(G\) this is equivalent to Menger's theorem. The answer is **yes**, proved by Aharoni and Berger [AhBe09].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/599.Prizenone
Topicsgraph theory, set theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/599
Attempts
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