Skip to content
Cairn Commons
X Sign in
Level A · Machine-checkable Hard Graph theory P-erdos-714

Erdős Problem #714

Is it true that ex(n; K_r,r) ≫ n^2-1/r?

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.

Start working on it Submit a claim Follow
Cite 
@misc{cairn-erdos-714,
  title        = {Erdős Problem #714},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-714}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims
0
Verified
0
Disputed
0
Refuted
0
On the literature board
0

Current state

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

The problem

The question

Is it true that

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«714». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_714 : answer(sorry) ↔
    ∀ r : ℕ, 2 ≤ r → ∃ c : ℝ, 0 < c ∧ ∀ᶠ n : ℕ in atTop,
      c * (n : ℝ) ^ ((2 : ℝ) - 1 / (r : ℝ)) ≤
        (extremalNumber n (completeBipartiteGraph (Fin r) (Fin r)) : ℝ)

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/714. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/714
  • [Br66] Brown, W. G., On graphs that do not contain a Thomsen graph. Canad. Math. Bull. (1966), 281-285.
  • [ERS66] Erdős, P. and Rényi, A. and Sós, V. T., On a problem of graph theory. Studia Sci. Math. Hungar. (1966), 215--235.
  • [KST54] Kövari, T. and Sós, V. T. and Turán, P., On a problem of K. Zarankiewicz. Colloq. Math. (1954), 50-57.

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.