Problem 1133

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claims/: The 2 claim pages of Problem 1133, one per claimant's result; the problem's standing derives from them.

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

For any $x_1,\ldots,x_n\in [-1,1]$ there exist $y_1,\ldots,y_n\in [-1,1]$ such that, if $P$ is a polynomial of degree $m<(1+\epsilon)n$ with $P(x_i)=y_i$ for at least $(1-\epsilon)n$ many $1\leq i\leq n$, then

$$\max_{x\in [-1,1]}\lvert P(x)\rvert >C.$$

Status. The site labels the problem OPEN (page last edited 31 December 2025; proof-claims tab accessed 2026-10-06). Two manuscripts claim the assertion in full, neither reviewed nor refereed: a note posted in the site's thread by Przemek Chojecki on 29 April 2026, produced with GPT-5.5 Pro as Chojecki wrote there and hosted at ulam.ai, which derives a finite obstruction from Beurling's interpolation-density theorem for the Bernstein space and plants it on short blocks of nodes (claim page), and the manuscript of Jia-Qi Yang submitted to the site's proof-claims tab on 2026-09-13 (using GPT-6, as the tab writes it), which claims to prove the obstruction for sign data at any pointwise tolerance with $\epsilon$ of optimal exponential order in $C$, and the sharp coefficient $\pi/2$ on Chebyshev–Lobatto grids (claim page). The site's commentary credits Erdős with a weaker statement, which he gives as Theorem 4 of [Er67, p. 72] and says he had stated without proof in an earlier paper: for every $C>0$ there is $\epsilon>0$ such that, for large $n$ and any $m=\lfloor(1+\epsilon)n\rfloor$ nodes in $[-1,1]$, some polynomial $P$ of degree $n$ has $\lvert P(x_i)\rvert\le1$ at every node and $\max\lvert P\rvert>C$; the assertion of the problem would imply it, and Erdős remarks in [Er67] that he could not prove the assertion even for $m=n$. This page records the claims without adopting them; the standing in the frontmatter follows from them.

Source. erdosproblems.com/1133, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1133, https://www.erdosproblems.com/1133.

References.

• [Er67] Erdős, P., Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems. Mathematica (Cluj) 10 (33) (1968), 65-73.

Formalization. Statement in formal-conjectures; the file states the problem and a weaker variant without proofs and names no formal proof. The community database at teorth/erdosproblems lists the statement as formalized since 2026-06-08 and the problem as unformalized otherwise.

Current assessment

The question (site formulation, page last edited 31 December 2025). For every $C>0$ there is $\epsilon>0$ such that, for all large $n$ and every $n$ nodes in $[-1,1]$, some labels in $[-1,1]$ force every polynomial of degree below $(1+\epsilon)n$ that matches at least $(1-\epsilon)n$ of them to exceed $C$ in absolute value somewhere on $[-1,1]$. OPEN. The site's commentary credits Erdős with the weaker statement, Theorem 4 of [Er67, p. 72], which he says he had stated without proof in an earlier paper, with $m=\lfloor(1+\epsilon)n\rfloor$ nodes and a polynomial of degree $n$ bounded by $1$ at every node, and his remark in [Er67] that he could not prove the assertion even for $m=n$. The statement fixes no dependence of $\epsilon$ on $C$.

Pending claims. Two full claims, neither reviewed nor refereed, and neither adopted. Chojecki 2026: a note posted in the site's thread on 29 April 2026, produced with GPT-5.5 Pro as the poster wrote, which derives from Beurling's interpolation-density theorem for the Bernstein space a finite forbidden label pattern on short blocks of nodes and plants it on more than $\epsilon n$ blocks after the substitution $x=\cos\theta$; it gives no rate for $\epsilon$ in $C$, and a same-day reply reporting a tool-assisted check is not a review. Yang 2026: a manuscript submitted to the proof-claims tab on 2026-09-13, using GPT-6 as the tab discloses, which claims the obstruction for sign data at any pointwise tolerance $\rho$ with $\epsilon$ of exponential order in $C/(1-\rho)$, shows that order optimal through interpolation at Chebyshev–Lobatto nodes, and identifies the sharp coefficient $\pi/2$ on the full Chebyshev–Lobatto grids, the sharp coefficient for arbitrary nodes being left open; it credits the earlier note for the angular reduction and grouping and takes its quantitative input from Olevskii and Ulanovskii. The derived standing is claimed with the claim value proved. Neither proof is compiled or reviewed in this wiki.

Formalization. The formal-conjectures file linked above states the problem and a weaker variant without proofs and names no formal proof; no Lean development of either claim is known.

Search scope (2026-10-07). The site's problem page, its thread with the three posts of 29 April and 13 September 2026, and its proof-claims tab with Yang's claim; the community database at teorth/erdosproblems; the formal-conjectures file at the pinned commit; the note at ulam.ai through its card and Yang's manuscript at its pinned commit.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• anon_2026_bernstein_density_proof_erdos_s_robust
• anon_2026_bernstein_density_proof_erdos_s_robust / lemma_3_2
• anon_2026_bernstein_density_proof_erdos_s_robust / lemma_4_1
• anon_2026_bernstein_density_proof_erdos_s_robust / proposition_3_1
• anon_2026_bernstein_density_proof_erdos_s_robust / theorem_1_1
• erdos_1967_problems_results_convergence_divergence_properties_lagrange
• erdos_1967_problems_results_convergence_divergence_properties_lagrange / conjecture_p72
• erdos_1967_problems_results_convergence_divergence_properties_lagrange / theorem_3
• erdos_1967_problems_results_convergence_divergence_properties_lagrange / theorem_4