Erdős Problem #1150

Is there some constant c > 0 such that, for all large enough n and all polynomials P of degree n with coefficients in -1, 1, max_|z|=1 |P(z)| > (1 + c) √(n)?

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-1150,
  title        = {Erdős Problem #1150},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1150}},
  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

Is there some constant $c > 0$ such that, for all large enough $n$ and all polynomials $P$ of degree $n$ with coefficients in ${- 1, 1}$ , $∣ z ∣ = 1 max ∣ P (z) ∣ > (1 + c) n ?$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1150». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1150 :
    answer(sorry) ↔ ∃ c > 0, ∀ᶠ n in Filter.atTop,
      ∀ P : ℂ[X],  (∀ i ≤ P.natDegree, P.coeff i = - 1 ∨ P.coeff i = 1) → P.natDegree = n →
        ⨆ z : Metric.sphere (0 : ℂ) 1, ‖P.eval (z : ℂ)‖ > (1 + c) * Real.sqrt n

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/1150. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1150

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.