115.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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 115

References:

erdosproblems.com/115
[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
[Er90] Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478.
[ErLe94] Erëmenko, A. and Lempert, L., An extremal problem for polynomials. Proc. Amer. Math. Soc. (1994), 191-193.
[Po59a] Pommerenke, Ch., On the derivative of a polynomial. Michigan Math. J. (1959), 373-375.

Imports (1)

FormalConjecturesUtil

theorem Erdos115.erdos_115 Documented declaration: erdos_115 sorry line 50 source
Source documentation
If $p(z)$ is a polynomial of degree $n$ such that $\{z : \lvert p(z)\rvert\leq 1\}$ is connected then is it true that $$\max_{\substack{z\in\mathbb{C}\\ \lvert p(z)\rvert\leq 1}} \lvert p'(z)\rvert \leq (\tfrac{1}{2}+o(1))n^2?$$
Eremenko and Lempert [ErLe94] have shown this is true, and in fact Chebyshev polynomials are the extreme examples.

depends on sorry what this means
Statement uses: Iff, True, Real, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, Polynomial, Complex, Polynomial.Monic, Eq, Polynomial.natDegree, IsConnected, Set.ofPred, LE.le, Norm.norm, Polynomial.eval, LinearMap, RingHom.id, Polynomial.derivative and 5 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
80 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_115 : answer(True) ↔
    ∀ ε > (0 : ℝ), ∀ᶠ n : ℕ in atTop, ∀ p : Polynomial ℂ, p.Monic → p.natDegree = n →
      IsConnected {z : ℂ | ‖p.eval z‖ ≤ 1} → ∀ z : ℂ, ‖p.eval z‖ ≤ 1 →
        ‖p.derivative.eval z‖ ≤ (1 / 2 + ε) * (n : ℝ) ^ 2 := by
…
```