Korsky: An improved lower bound for the de Bruijn–Erdős consecutive gap problem

../

theorem_1_1: For every r at least 2 and every sequence of distinct points on the circle, the upper limit of the ratio of the largest to the smallest r-span is at least 1 + r/(r^2 − 1), so 5/3 for r = 2; a fixed-r improvement of the 1949 bound 1 + 1/r.

---

Samuel Korsky, An improved lower bound for the de Bruijn--Erdős consecutive gap problem, arXiv:2605.30959v1 (29 May 2026), math.CO, 8 pages, manuscript dated June 1, 2026. Unrefereed; no journal reference on arXiv. Suggested key [Ko26a].

Retained artifact. The folder-name PDF is arXiv v1, retrieved from https://arxiv.org/pdf/2605.30959v1; 268,639 bytes; the arXiv v1 LaTeX source and HTML rendering were fetched too, and the arXiv metadata sits in the .arxiv/ sidecar. The text layer was read. The arXiv record (https://arxiv.org/abs/2605.30959, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Read status. Claims checked for Theorem 1.1 (p. 2), read clause by clause; the proof (Sections 2--5, pp. 2--7) was read for its structure only and not checked. Unrefereed; nothing here is independently reviewed.

Overview

For a sequence of distinct points on $\mathbb T=\mathbb R/\mathbb Z$ the first $n$ points cut $n$ intervals, and $M_n^{(r)}$, $m_n^{(r)}$ are the maximum and the minimum, over runs of $r$ consecutive intervals, of the summed lengths. The paper recalls the 1949 bound $\limsup_nM_n^{(r)}/m_n^{(r)}\ge1+1/r$ ((5.7)), sharp for $r=1$, and the upper construction $1+O(\log r/r)$ of Clément and Steinerberger. Theorem 1.1 (p. 2): whatever the integer $r\ge2$ and the sequence of distinct points, $\limsup_nM_n^{(r)}/m_n^{(r)}\ge1+r/(r^2-1)$; since $r/(r^2-1)=1/r+1/(r(r^2-1))$ this strictly improves $1+1/r$ for each $r\ge2$, and for $r=2$ gives $5/3$ in place of $3/2$.

The argument (Sections 2--5) rests on locality: a new point splits a single interval, so only the $r$-blocks near it change. $M_n$ is nonincreasing (Lemma 2.1, p. 3) and $m_n\le r/n\le M_n$. If the ratio stays below a fixed $\rho$, a "slow" split, one that does not lower $M_n$ by the factor $1-\eta$, creates a protected block of $2r$ intervals none of which can be split again until $M_n$ has fallen by the factor $\beta=(r-1)(\rho-1+\eta)$ (Lemma 3.1, p. 3). Counting protected blocks over one multiplicative epoch shows that the time $N^+$ at which $M$ first falls below $\beta M_N$ satisfies $N^+\le(1+1/r)N+C$ (Proposition 4.1, p. 5). Iterating, $M_{N_j}\ge r/N_j$ decays no faster than $(r/(r+1))^j$ while by construction it decays at least like $\beta^j$; choosing $\rho<1+r/(r^2-1)$ and $\eta$ small makes $\beta<r/(r+1)$, a contradiction (Section 5, p. 7). Section 6 (p. 8) notes that the improvement is far from the logarithmic scale of the upper construction and records Conjecture 6.1, that $\limsup_nM_n^{(2)}/m_n^{(2)}\ge2$ for every sequence of distinct points.

Relation to Problem 1221: a fixed-$r$ improvement of the third bound; it does not address the growth of $r(\mu_r-1)$ as $r\to\infty$. The author's later preprint claims that growth (Theorem 1.1 there, claimed and unreviewed) and cites this note as the starting point of that work; the site's proof-claim comment by the author says this note involved essentially no AI use.

Bears on. Problem 1221: fixed-$r$ progress on the third constant, $\mu_r\ge1+r/(r^2-1)$ for $r\ge2$ over sequences of distinct points; unrefereed.