Smallest dimension where Borsuk's conjecture fails
the smallest with a known counterexample to Borsuk's conjecture in
- steps
- 10
- by AI
- 1
- since
- 1993
The line is the frontier over time, falling as the bound comes down: lower is better here. Filled dots are steps that moved it; muted dots are results that did not. Orange dots are catalog entries, results with AI in the loop. Hollow dots are candidates under review and never move the line.Grey dots along the top edge are results from before the quantity had a number, placed at the worst end because they have no value on this axis. Dots that would overlap are nudged sideways a few pixels. Hover a dot for its value and attribution.
About this frontier
Borsuk asked in 1933 whether every bounded set in can be partitioned into subsets of strictly smaller diameter. True for ; false in general, as Kahn and Kalai showed in 1993 with a counterexample in dimension 1325. Since then the question has been where it first fails, and the race has been to lower the dimension of an explicit counterexample: 946, 561, 560, 323, 321, 298, then Bondarenko's two-distance construction at 65 in 2013, Jenrich and Brouwer's 64, and 63 in 2026. The first failing dimension is open everywhere in , so the finish line is unknown; this frontier tracks the ceiling on it.
Every step, newest first
| Date | Value | Who | Model | Status | Source |
|---|---|---|---|---|---|
| 26 May 2026 | best A 321-point subset of R^63 whose smaller-diameter subsets have at most 5 points, so at least 65 > 64 parts are needed: a 320-point rank-63 piece of the G2(4) set plus one scaled projected point. The repository's exact verifier was rerun here on 12 August. Found again independently by Nicholas Konz with Claude in August 2026. On Tao's ledger as the current best. | Max Grinsztajn, with GPT-5.5 Pro | GPT-5.5 Pro | AI step site-confirmed | entry |
| Aug 2013 | A 352-point two-distance subset of Bondarenko's configuration, needing at least 71 parts, three months after his paper. Electronic Journal of Combinatorics 21 (2014); dated to the arXiv posting. | Thomas Jenrich and Andries E. Brouwer | – | historical | source ↗ |
| May 2013 | The big drop: a 416-point two-distance set on the sphere in R^65, from the G2(4) strongly regular graph, that cannot be split into 83 parts of smaller diameter. Discrete and Computational Geometry 51 (2014); dated here to the May 2013 arXiv posting. | Andriy Bondarenko | – | historical | source ↗ |
| 2003 | Discrete Mathematics 270 (2003). The record for ten years. | Aicke Hinrichs and Christian Richter | – | historical | source ↗ |
| Mar 2002 | Counterexamples in dimensions 321 and 322, one month after Hinrichs. | Oleg Pikhurko | – | historical | source ↗ |
| Jan 2002 | Discrete Mathematics 243 (2002). A construction from spherical codes, which is the idea the next three steps refine. | Aicke Hinrichs | – | historical | source ↗ |
| 2000 | Beitraege zur Algebra und Geometrie 41 (2000), 417-423, one dimension below Raigorodskii. The EMIS archive refused automated access from here; the value is as cited by Bondarenko and by Tao's ledger. | Bernulf Weissbach | – | historical | source ↗ |
| 1997 | Russian Mathematical Surveys 52 (1997), a two-page note. | Andrei M. Raigorodskii | – | historical | source ↗ |
| 1994 | Jerusalem Combinatorics '93, Contemporary Mathematics 178. A sharpening of the Kahn-Kalai construction; Nilli is a pseudonym of Noga Alon. | A. Nilli | – | historical | source ↗ |
| 1993 | The disproof. Kahn and Kalai showed the Borsuk number grows at least like exp(c sqrt n), so the conjecture fails in every sufficiently large dimension, and exhibited the failure explicitly at n = 1325. Bulletin of the AMS 29 (1993). Everything below is the race to bring that number down. | Jeff Kahn and Gil Kalai | – | historical | source ↗ |
Every row is on Tao's optimisation-problems ledger (constants/28a.md), read here; each reference was then checked against the linked paper's own metadata, and Bondarenko's paper cites the 1994 to 2003 values as well. The two 2013 rows are dated to their arXiv postings, not their 2014 journal issues. Weissbach's paper is linked at the EMIS journal archive, which refused automated access from here; that row rests on Bondarenko's and Tao's citations. The 2026 row is published rather than a candidate because the entry is site-confirmed: its exact verifier was rerun here on 12 August.