Erdős Problem #1133

Let C>0. There exists ε>0 such that if n is sufficiently large the following holds. For any x_1,…,x_n∈ [-1,1] there exist y_1,…,y_n∈ [-1,1] such that, if P is a polynomial of degree m<(1+ε)n with P(x_i)=y_i for at least (1-ε)n many 1≤ i≤ n, then max_x∈ [-1,1]lvert P(x)rvert >C.

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-1133,
  title        = {Erdős Problem #1133},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1133}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $C > 0$ . There exists $ϵ > 0$ such that if $n$ is sufficiently large the following holds.

For any $x_{1}, \dots, x_{n} \in [- 1, 1]$ there exist $y_{1}, \dots, y_{n} \in [- 1, 1]$ such that, if $P$ is a polynomial of degree $m < (1 + ϵ) n$ with $P (x_{i}) = y_{i}$ for at least $(1 - ϵ) n$ many $1 \leq i \leq n$ , then $x \in [- 1, 1] max ∣ P (x)∣ > C .$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1133». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1133 :
    answer(sorry) ↔
    ∀ C > (0 : ℝ), ∃ ε > (0 : ℝ), ∀ᶠ n : ℕ in atTop,
      ∀ x : Fin n → Icc (-1 : ℝ) 1,
        ∃ y : Fin n → Icc (-1 : ℝ) 1,
          ∀ P : Polynomial ℝ,
            (P.natDegree : ℝ) < (1 + ε) * (n : ℝ) →
            ((Finset.univ.filter (fun i ↦ P.eval (x i : ℝ) = (y i : ℝ))).card : ℝ) ≥
              (1 - ε) * (n : ℝ) →
            ∃ z ∈ Icc (-1 : ℝ) 1, |P.eval z| > C

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/1133. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1133
[Er67] Erdős, P., Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems. Mathematica (Cluj) (1967), 65-73.

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.