From 24eebca62f9dfa228271dfa203f7f729aab1be59 Mon Sep 17 00:00:00 2001 From: Taksh Date: Sun, 6 Sep 2026 22:37:09 +0530 Subject: [PATCH] =?UTF-8?q?Record=20the=20Pesenti=E2=80=93Vladu=20v2=20err?= =?UTF-8?q?atum=20for=20the=20Spencer=20constant.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit The 3.674 upper bound is the withdrawn 3.7 in Theorem 4.5; v2 states 4.1. --- README.md | 4 +++- constants/10c.md | 5 ++++- 2 files changed, 7 insertions(+), 2 deletions(-) diff --git a/README.md b/README.md index c2c630f..6aa3821 100644 --- a/README.md +++ b/README.md @@ -33,7 +33,7 @@ Bounds for which the level of available verification is currently at minimal lev | [9](https://teorth.github.io/optimizationproblems/constants/9a.html) | Shannon capacity of the 7-cycle | 3.2578 | 3.3177 | | [10a](https://teorth.github.io/optimizationproblems/constants/10a.html) | The real Grothendieck constant | $\frac{6\pi}{11}\approx 1.71360$ | $\frac{\pi}{2\log(1+\sqrt{2})} - 10^{-4} \approx 1.78211$ | | [10b](https://teorth.github.io/optimizationproblems/constants/10b.html) | The complex Grothendieck constant | 1.338 | 1.40491 | -| [10c](https://teorth.github.io/optimizationproblems/constants/10c.html) | Spencer discrepancy constant (“six standard deviations suffice”) | 1.697749 | 3.674235 (3.65*) | +| [10c](https://teorth.github.io/optimizationproblems/constants/10c.html) | Spencer discrepancy constant (“six standard deviations suffice”) | 1.697749 | 4.1 (3.65*) | | [11a](https://teorth.github.io/optimizationproblems/constants/11a.html) | $L^1$ Poincaré constant on the Hamming cube | $\sqrt{\pi/2} \approx 1.2533$ | $\pi/2 - 0.00013 \approx 1.5707$ | | [11b](https://teorth.github.io/optimizationproblems/constants/11b.html) | Critical exponent for isoperimetric inequality on the Hamming cube | 0.5 | 0.5 | | [12](https://teorth.github.io/optimizationproblems/constants/12a.html) | The Beardwood–Halton–Hammersley constant | 0.6277 | 0.90304 | @@ -159,6 +159,8 @@ Bounds for which the level of available verification is currently at minimal lev - [43](https://teorth.github.io/optimizationproblems/constants/43a.html) **improved lower bound (unverified):** $C_{43} \geq 0.860*$ (exact $43/50$; certificate-layer result conditional on the lemma set of [KHSHGW2026](https://arxiv.org/abs/2601.22365)) by [J. Savva](https://doi.org/10.5281/zenodo.22223485), 1 Sep 2026. - [88a](https://teorth.github.io/optimizationproblems/constants/88a.html) **improved upper bound:** $C_{88a} \leq 186$ via $\mathrm{DHL}[40,2]$, by [OpenAI](https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf), 30 Aug 2026, with a Lean 4 formalization conditional on three declared axioms. +- [10c](https://teorth.github.io/optimizationproblems/constants/10c.html) **upper bound correction:** $C_{10c}\le 4.1$ by [Pesenti–Vladu](https://arxiv.org/abs/2211.05509) v2 (14 Apr 2026), replacing the withdrawn $3\sqrt{3/2}$ constant in Theorem 4.5. + ## Maintainers This site is maintained by Damek Davis, Paata Ivanisvili and Terence Tao. diff --git a/constants/10c.md b/constants/10c.md index aa8e7e5..633794a 100644 --- a/constants/10c.md +++ b/constants/10c.md @@ -36,7 +36,8 @@ $$ | $5.32$ | [Spe1985] | Usually reported as $6$. The celebrated “six standard deviations suffice” theorem of Spencer; also applies to rectangular matrices or set systems. | | $5.199$ | [Bel2013] | Re-optimizes Spencer’s method. | | $3.65$ (unpublished) | Schmidt [Bel2013] | Some of the computations are given only as a personal communication. | -| $3\sqrt{3/2}\approx 3.674235$ | [PV2022] | Also gives an algorithmic version. | +| $3\sqrt{3/2}\approx 3.674235$ | [PV2022] | Withdrawn. arXiv v2 (14 Apr 2026) comments: “the constant in Theorem 4.5 is corrected from 3.7 to 4.1”. | +| $4.1$ | [PV2022] | v2 Theorem 4.5: $\lVert Ax\rVert_\infty\leqslant 4.1\sqrt{n}+O(1)$. Algorithmic. | ## Known lower bounds @@ -360,3 +361,5 @@ The centres $T_i$ were found by a greedy set-cover search for a small radius-$5$ ## Contribution notes ChatGPT Pro was used to generate an initial version of this page. + +The v2 erratum on Theorem 4.5 was checked against the arXiv comments field of [2211.05509](https://arxiv.org/abs/2211.05509) and against the statement of Theorem 4.5 in the v2 html. AI assistance was used to draft the wording.