Erdős Problem #572 — open problem in graph theory

Erdős Problem #572

Show that for k≥ 3 ex(n;C_2k)≫ n^1+1/k. This problem is #46 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-572,
  title        = {Erdős Problem #572},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-572}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Show that for $k \geq 3$ $ex (n; C_{2 k}) ≫ n^{1 + \frac{1}{k}} .$

This problem is #46 in Extremal Graph Theory in the graphs problem collection.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«572».

theorem erdos_572 (k : ℕ) (hk : 3 ≤ k) :
    ∃ c > (0 : ℝ), ∀ᶠ (n : ℕ) in atTop,
      c * (n : ℝ) ^ (1 + 1 / (k : ℝ)) ≤
        (SimpleGraph.extremalNumber n (SimpleGraph.cycleGraph (2 * k)) : 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/572. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/572
[Er64c] Erdős, P., Extremal problems in graph theory. Theory of Graphs and its Applications (1964), 29-36.
[BoSi74] Bondy, J. A. and Simonovits, M., Cycles of even length in graphs. J. Combin. Theory Ser. B (1974), 97-105.
[LUW95] Lazebnik, F., Ustimenko, V. A. and Woldar, A. J., A new series of dense graphs of high girth. Bull. Amer. Math. Soc. (N.S.) (1995), 73-79.

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.