572.lean`

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L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 572

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.

Imports (1)

FormalConjecturesUtil

theorem Erdos572.erdos_572 Show that for $k\geq 3$ $$\mathrm{ex}(n;C_{2k})\gg n^{1+\frac{1}{k}}.$$ This problem is #46 in Extremal Graph Theory in the graphs problem collection sorry line 46 source
Source documentation
Show that for $k\geq 3$ $$\mathrm{ex}(n;C_{2k})\gg n^{1+\frac{1}{k}}.$$
This problem is #46 in Extremal Graph Theory in the graphs problem collection.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, Real, And, GT.gt, Filter.Eventually, HMul.hMul, HPow.hPow, HAdd.hAdd, HDiv.hDiv, SimpleGraph.extremalNumber, Fin, SimpleGraph.cycleGraph, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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) := by
…
```
Mentions: c