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Source. S. Korsky, An improved lower bound for the de Bruijn--Erdős consecutive gap problem, arXiv:2605.30959v1 (29 May 2026), Theorem 1.1 (p. 2) and its proof in Section 5 (p. 7) of the retained PDF, read in the canonical conversion and checked against the text layer; held by its library card, Korsky 2026, improved lower bound, with the result page Theorem 1.1. The inputs are reconstructed on the Lemma 3.1 page (with Lemma 2.1 and the mean identity) and the Proposition 4.1 page.

Standing. Author-recorded reconstruction; not an independent review; changes no status and assigns no tier. The source is an unrefereed preprint; the argument is self-contained and uses no external theorem.

Definitions

Let $x_1,x_2,\ldots$ be distinct points of $\mathbb T=\mathbb R/\mathbb Z$; the first $n$ points cut the circle into $n$ gaps. For a fixed integer $r\ge2$, $M_n=M_n^{(r)}$ and $m_n=m_n^{(r)}$ are the largest and smallest sums of $r$ cyclically consecutive gaps at time $n$, and $R_n=M_n/m_n$.

Statement (Theorem 1.1, p. 2)

For every integer $r\ge2$ and every sequence of distinct points on $\mathbb T$,

$$\limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}}\ \ge\ 1+\frac r{r^2-1}.$$

Since $r/(r^2-1)=1/r+1/(r(r^2-1))$, this exceeds the 1949 bound $1+1/r$ of (5.7) for every $r\ge2$; for $r=2$ it gives $5/3$.

Proof

Suppose, for a contradiction, that $\limsup_nR_n<1+r/(r^2-1)$. Choose $\rho$ with

$$\limsup_{n\to\infty}R_n<\rho<1+\frac r{r^2-1},$$

so that $R_n\le\rho$ for all $n\ge N_1$, for some $N_1$. Since $\rho-1<r/(r^2-1)=\frac r{(r-1)(r+1)}$, we may choose $\eta\in(0,1)$ so small that

$$\beta=(r-1)(\rho-1+\eta)\ <\ \frac r{r+1};$$

then also $0<\beta<1$, as Proposition 4.1 requires.

The epoch sequence. Take $N_0\ge\max(N_1,2r)$ large enough that Proposition 4.1 applies to every $N\ge N_0$, and define recursively

$$N_{j+1}=N_j^+ ,$$

the first time after $N_j$ with $M_{N_{j+1}}\le\beta M_{N_j}$. By Proposition 4.1 there is $C_0=C(r,\eta,\rho)$ with

$$N_{j+1}\ \le\ \Bigl(1+\frac1r\Bigr)N_j+C_0\qquad(j\ge0).$$

Adding $rC_0$ to both sides gives $N_{j+1}+rC_0\le(1+1/r)(N_j+rC_0)$, so by induction

$$N_j\ \le\ N_j+rC_0\ \le\ C_1\Bigl(1+\frac1r\Bigr)^j ,\qquad C_1=N_0+rC_0 .$$

Two incompatible decay rates. The mean identity $M_n\ge r/n$ gives

$$M_{N_j}\ \ge\ \frac r{N_j}\ \ge\ \frac r{C_1}\Bigl(\frac r{r+1}\Bigr)^j =C_2\Bigl(\frac r{r+1}\Bigr)^j,\qquad C_2=\frac r{C_1}>0 .$$

By construction $M_{N_{j}}\le\beta M_{N_{j-1}}$ for every $j\ge1$, so

$$M_{N_j}\ \le\ \beta^jM_{N_0}.$$

Together, $C_2(r/(r+1))^j\le\beta^jM_{N_0}$, that is,

$$\Bigl(\frac{r/(r+1)}{\beta}\Bigr)^j\ \le\ \frac{M_{N_0}}{C_2}\qquad(j\ge0).$$

The base exceeds $1$ because $\beta<r/(r+1)$, so the left side tends to infinity with $j$, which is impossible. Hence $\limsup_nR_n\ge1+r/(r^2-1)$.

Source notes

• The constants $N_0,C_0,C_1,C_2$ depend on $r$, $\rho$, $\eta$ and the sequence; the theorem is a fixed-$r$ statement and asserts no uniformity in $r$.
• The source's Section 6 remarks that the improvement is far from the logarithmic scale of the upper construction and records Conjecture 6.1, $\limsup_nM_n^{(2)}/m_n^{(2)}\ge2$; neither is reconstructed.
• Distinctness of the points is used only to make every gap, hence every $m_n$, positive; the splitting dynamics is otherwise the same as in the 1949 note.

Reading addressed

The theorem concerns the third constant $\mu_r=\inf_X\mu_r(X)$ over sequences $X$ of distinct points, the same under the literal and the mean-normalized readings of Problem 1221. It gives $r(\mu_r-1)\ge1+1/(r^2-1)$ for each $r\ge2$, which stays bounded, so it does not bear on whether $r(\mu_r-1)\to\infty$; that growth is the ratio part of Korsky's 2026 preprint.