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Level A · Machine-checkable Hard Graph theory P-erdos-23

Erdős Problem #23

Can every triangle-free graph on 5n vertices be made bipartite by deleting at most n^2 edges?

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-23,
  title        = {Erdős Problem #23},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-23}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Can every triangle-free graph on vertices be made bipartite by deleting at most edges?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«23». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_23 : answer(sorry) ↔
    ∀ (n : ℕ) (V : Type) [Fintype V], Fintype.card V = 5 * n →
      ∀ (G : SimpleGraph V), G.CliqueFree 3 →
        ∃ (H : SimpleGraph V),
          H ≤ G ∧ H.IsBipartite ∧ (G.edgeFinset \ H.edgeFinset).card ≤ n^2

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/23. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

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.