Erdős Problem #65 — open problem in graph theory

Erdős Problem #65

Is the sum Σ1/a_i minimised when G is a complete bipartite graph? This problem is #65 in Extremal Graph Theory in the graphs problem collection.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-65,
  title        = {Erdős Problem #65},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-65}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Is the sum $\sum \frac{1}{a _{i}}$ minimised when $G$ is a complete bipartite graph?

This problem is #65 in Extremal Graph Theory in the graphs problem collection.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«65». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_65.parts.ii : answer(sorry) ↔
    ∀ (k : ℝ) (hk : 0 < k),
      ∀ (n : ℕ) (V : Type) [Fintype V] (G : SimpleGraph V),
        0 < n →
        Fintype.card V = n →
        (G.edgeSet.ncard : ℝ) = k * n →
        ∀ (A B : Type) [Fintype A] [Fintype B],
          Fintype.card (A ⊕ B) = n →
          ((completeBipartiteGraph A B).edgeSet.ncard : ℝ) = k * n →
          (∑ᶠ a ∈ (completeBipartiteGraph A B).cycleLengths, (1 : ℝ) / a) ≤
            (∑ᶠ a ∈ G.cycleLengths, (1 : ℝ) / a)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/65. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/65
[GKS84] Gyárfás, A., Komlós, J. and Szemerédi, E., On the distribution of cycle lengths in graphs. J. Graph Theory (1984), 441-462.
[LiMo20] Liu, H. and Montgomery, R., A solution to Erdős and Hajnal's odd cycle problem. arXiv:2010.15802 (2020).

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.