Problem 1221

Let $a=(a_1,\ldots,a_n)$ be points, ordered consecutively, around the circle (normalised to have circumference $1$ ). Let

M_r(a)=\max_r \lvert a_{i+r}-a_i\rvert

and

m_r(a)=\min_r \lvert a_{i+r}-a_i\rvert,

where the indices are interpreted in a cyclic fashion and the distance is around the circle (in other words, the maximum and minimum distance between $r$ -consecutive points). Let

\Lambda_r=\inf_a \limsup_n n M_r(a_1,\ldots,a_n),

\lambda_r = \sup_a \liminf_n nm_r(a_1,\ldots,a_n),

and

\mu_r=\inf_a \limsup_n \frac{M_r(a)}{m_r(a)},

where the outer $\inf/\sup$ is over all infinite sequences $a=(a_1,\ldots)$ . Is it true that $r(\Lambda_r-1)$ , $r(1-\lambda_r)$ , and $r(\mu_r-1)$ tend to infinity as $r\to \infty$ ?

Second statement

Let $a=(a_1,\ldots,a_n)$ be points, ordered consecutively, around the circle (normalised to have circumference $1$ ). Let

M_r(a)=\max_r \lvert a_{i+r}-a_i\rvert

and

m_r(a)=\min_r \lvert a_{i+r}-a_i\rvert,

where the indices are interpreted in a cyclic fashion and the distance is around the circle (in other words, the maximum and minimum distance between $r$ -consecutive points). Let

\Lambda_r=\inf_a \limsup_n n M_r(a_1,\ldots,a_n),

\lambda_r = \sup_a \liminf_n nm_r(a_1,\ldots,a_n),

and

\mu_r=\inf_a \limsup_n \frac{M_r(a)}{m_r(a)},

where the outer $\inf/\sup$ is over all infinite sequences $a=(a_1,\ldots)$ . Is it true that $\Lambda_r-r$ , $r-\lambda_r$ , and $r(\mu_r-1)$ tend to infinity as $r\to \infty$ ?

Source: erdosproblems.com/1221

Analysis

Status

Open

No claim settles this problem.

The site's label is OPEN. A preprint [Ko26b] (arXiv:2609.07196v2, 9 September 2026) claims all three parts of the corrected Statement for sequences of distinct points, and is registered on the site as a full proof claim (submitted 2026-09-08); it is recorded as the pending partial claim Korsky 2026, since its constants run over sequences of distinct points only, and the frontmatter standing is open. No acceptance evidence was found in a search: the preprint is unrefereed, the site shows OPEN, the community database keeps open (2026-09-18), and no independent review exists.

Notes

With the site's definitions, $M_r$ and $m_r$ are the largest and smallest sums of $r$ consecutive gaps (the "distance between $r$ -consecutive points" is the forward arc through the $r-1$ intervening points), whose mean is $r/n$ , so $nM_r\ge r\ge nm_r$ . The site's first two expressions then fail as a question. Section 3 of [dBEr49] gives $\Lambda_r\ge1/\log(1+1/r)>r$ , so $r(\Lambda_r-1)\ge r(r-1)$ and the first tends to infinity trivially. A sum of $r$ consecutive gaps is at least $r$ times the smallest gap, so $\lambda_r\ge r\lambda_1=r/\log4>1$ for $r\ge2$ , and $r(1-\lambda_r)\le r(1-r/\log4)$ is already negative at $r=2$ and tends to $-\infty$ ; read as printed, the question has the answer no. The library's conjecture page writes the argument out. The change replaces " $r(\Lambda_r-1)$ , $r(1-\lambda_r)$ " by " $\Lambda_r-r$ , $r-\lambda_r$ ", which normalizes $\Lambda_r$ and $\lambda_r$ by the mean span: $r(\Lambda_r/r-1)=\Lambda_r-r$ and $r(1-\lambda_r/r)=r-\lambda_r$ . The third expression needs no normalization and is unchanged. The evidence is the posers' own words in [dBEr49]. The introduction (p. 14) says: "All we can prove is that $\mu_r\geq 1+1/r$ (and analogous inequalities for $\Lambda_r$ and $\lambda_r$ ); we conjecture that $r(\mu_r-1)$ is unbounded." Section 6 (p. 17) opens: "The inequalities (3.3), (4.3) and (5.7) are probably not best possible if $r\geq2$ ." It then conjectures that the three expressions tend to infinity. Normalized by $r$ , the three bounds are analogous, of the form $1\pm c/r$ , and give exactly the bounded lower bounds $\Lambda_r-r\ge\tfrac12-\tfrac1{12r}+O(r^{-2})$ , $r-\lambda_r\ge\tfrac12-\tfrac5{12r}+O(r^{-2})$ and $r(\mu_r-1)\ge1$ that the conjecture says can be improved to growth. With the printed normalization, (3.3) would already prove the first part, and the second part would be false, so the posers' words are true only of the normalized form. The form is the one Korsky [Ko26b, p. 2] prints when he restates the conjecture of [dBEr49, p. 17], $\bar A_r-r\to\infty$ , $r-\underline A_r\to\infty$ , $r(\mu_r-1)\to\infty$ , apart from the hypotheses of his theorem (his constants are taken over sequences of distinct points, a restriction the correction does not adopt; see Formulation). The form is taken from these two sources, not from the results that bear on it. The defect is already in the source: Section 6 of [dBEr49] prints $r(\Lambda_r-1)$ and $r(1-\lambda_r)$ with the same definitions, and the site reproduces it; the site's page shows the community database's note that the original source is ambiguous. The failure was reported by the GitHub user OutBlade as issue 414 of the community database (teorth/erdosproblems#414, 11 September 2026), with a note at OutBlade/erdos-notes that proposes the missing factor of $r$ ; both say the observation was found and checked with Claude (Anthropic). The database's pull request 416, merged 18 September 2026, marks the entry "ambiguous statement". This is the one result about the site's wording; it is credited here and counts for nothing. Unread: the Brethouwer thesis (TU Delft 2024), which [Ko26b] and [ClSt25] cite for the third part.

dBEr49: de Bruijn, N. G. and Erdős, P., Sequences of points on a circle. Nederl. Akad. Wetensch., Proc. 52 (1949), 14--17 = Indagationes Math. 11 (1949), 46--49 (communicated 18 December 1948; MR 33331). Definitions and the $r=1$ values, Sections 1--2, pp. 14--15; the lower bound for $\Lambda_r$ , Section 3, p. 15 (unnumbered on the page, cited as (3.3) in Section 6; footnote 2 on p. 15 credits the Section 4 proof as found independently by van Aardenne-Ehrenfest); the upper bound (4.3) for $\lambda_r$ , p. 16; (5.1) and (5.7) for $\mu_r$ , pp. 16--17; the conjecture, Section 6, p. 17. The publisher's version of record is on the TU/e research portal (a cover sheet and four image-only article pages). Library home: Debruijn and Erdős 1949.
Ko26b: Korsky, S., A resolution of the de Bruijn–Erdős consecutive-gap problem. arXiv:2609.07196 (v1 7 September 2026, the ratio part only; v2 9 September 2026, all three parts; 16 pages; math.CO). Theorem 1.1, p. 2. The site's proof-claim link is the same manuscript (dated September 8, 2026) on Google Drive. Unrefereed; the paper's acknowledgments (p. 15) state that GPT Astra was used to locate Larcher's bound and to complete, audit and review the argument, with the author checking it independently. Library home: Korsky 2026.
Ko26a: Korsky, S., An improved lower bound for the de Bruijn–Erdős consecutive gap problem. arXiv:2605.30959v1 (29 May 2026; 8 pages). Theorem 1.1, p. 2: $\mu_r\ge1+r/(r^2-1)$ for $r\ge2$ . Unrefereed. Library home: Korsky 2026.
ClSt25: Clément, F. and Steinerberger, S., Balanced stick breaking. arXiv:2511.14637v1 (18 November 2025; 12 pages). Theorem 2, p. 2: $\mu_r\le1+c\log r/r$ for every $r\ge2$ . Unrefereed. Library home: Clément and Steinerberger 2025.
Be26: Bevan, D., On balancing consecutive slices of cake. arXiv:2607.00775v2 (7 July 2026; 8 pages). Upper bounds on $\inf_a\mu_r(a)$ for small $r$ from a family of sequences whose $\mu_r(a)$ is computable. Known from its abstract; lead. - Leads: DeLeo, J., Henderschedt, O. and Wells, C., A finite victory over de Bruijn–Erdős in interval discrepancy, arXiv:2605.29166 (the finite $r=1$ splitting problem); Brethouwer, J.-T. F., Ph.D. thesis, TU Delft 2024, doi 10.4233/uuid:617757b7-32d2-4bac-8be3-5a6a13e6e710, Section 3.3.1, Question 3 (whether $\mu_r-1\gg\log r/r$ ), as cited by [Ko26b] and [ClSt25]; van Aardenne-Ehrenfest, Proc. 48 (1945), 266--271 = Indag. Math. 7 (1946), 71--76, the "just distributions" theorem that [dBEr49] says would follow from the third part; a second Bevan preprint on two-slice portions, listed by Semantic Scholar as citing [Ko26a] and not located on arXiv by title.

Problem 1221

Second statement

Status

Notes

Proof claims

Discussion