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Borsuk's Conjecture

Borsuk's conjecture (Borsuk 1932) asserts that every bounded set of n-dimensional Euclidean space of generalized diameter 1 can be partitioned into n+1 pieces of strictly smaller generalized diameter. It is true for n=2 and 3 and for sets with smooth boundary. However, the number of pieces required in the worst case grows at least exponentially in sqrt(n), so the conjecture is false in sufficiently high dimensions.

Kahn and Kalai (1993) found a counterexample in dimension 1326, Nilli (1994) a counterexample in dimension 946. Hinrichs and Richter (2003) showed that the conjecture is false for all n>297.

Bondarenko (2014) reduced the dimension of a counterexample to 65, and Jenrich and Brouwer (2014) reduced it to 64. With assistance from GPT-5.5 Pro, Grinsztajn (2026) supplied a construction in dimension 63 with 321 points such that a subset of smaller generalized diameter has at most five points. At least [321/5]=65 pieces are therefore required, where [x] is the ceiling function, exceeding 63+1. This lowers the smallest dimension with a known finite set counterexample to 63.

OpenAI (2026, p. 2) announced a nine-dimensional counterexample consisting of the compact set

X={uu^T:u in R^4, ||u||=1}.

These projection matrices have matrix rank 1 and lie in the affine space of real symmetric matrices of order 4 with matrix trace 1, which has dimension 9. With distance measured by the Frobenius norm, X has generalized diameter sqrt(2) and cannot be covered by ten subsets of strictly smaller generalized diameter. This construction uses the full infinite set and does not supply a finite set counterexample in dimension 9 (Kalai 2026).

The release attributes the argument to an internal OpenAI model and supplies a Lean formalization. MathVet (2026) reports successful independent kernel checking of the formalized theorem. Its accompanying statement review is model-assisted and remains pre-referee.

The stronger quantitative Gale's conjecture asks whether the four pieces in three dimensions can always have generalized diameter at most sqrt((3+sqrt(3))/6) times that of the original set. Voronov (2026) refuted this strengthening with 101 points. This does not contradict the three-dimensional case of Borsuk's conjecture.

The Borsuk number b_d is the least integer m such that every bounded set of generalized diameter 1 in R^d can be partitioned into m subsets of strictly smaller generalized diameter. The vertices of a regular simplex show that b_d>=d+1. Borsuk's conjecture is the assertion b_d=d+1. The related Grünbaum's equal-diameter covering problem requires a cover by closed balls having the original generalized diameter.

For dimension four, Arman et al. (2026) proved that every bounded set of diameter 1 in R^4 can be covered by seven balls of a common diameter strictly less than 0.9999. Their result gives the four-dimensional Borsuk number bound 5<=b_4<=7. The upper bound uses an exact rational interval certificate. Arman et al. (2026) report AI assistance in finding the circumradius approach used in the proof.

Tolmachev (2026) improved the four-dimensional bound to 5<=b_4<=6. For dimensions 4 through 19, his upper bounds are, respectively, 6, 12, 31, 45, 89, 157, 289, 625, 1245, 2313, 4977, 7745, 13729, 24337, 48673, and 97345. In particular, b_4=5 remains open. For n>=9 and nonnegative integers a, b, and c with n-1=7a+8b+c, he obtained

b_n<=1+88^a156^b2^c.

These estimates concern worst-case partitions into sets of strictly smaller generalized diameter. Their computer-assisted parts give bounds in dimensions 4 through 6. Tolmachev (2026) reports that GPT-6 Astra and Claude Fable 5.2 assisted in obtaining the results.

See also

Generalized Diameter, Keller's Conjecture, Lebesgue Minimal Problem

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References

Arman, A.; Bondarenko, A.; Prymak, A.; and Radchenko, D. "Borsuk Number in Four Dimensions Is at Most Seven." 29 Oct 2026. https://arxiv.org/abs/2610.39078.Bondarenko, A. "On Borsuk's Conjecture for Two-Distance Sets." Discrete Comput. Geom. 51, 509-515, 2014. https://doi.org/10.1007/s00454-014-9579-4.Borsuk, K. "Über die Zerlegung einer Euklidischen n-dimensionalen Vollkugel in n Mengen." Verh. Internat. Math.-Kongr. Zürich 2, 192, 1932.Borsuk, K. "Drei Sätze über die n-dimensionale euklidische Sphäre." Fund. Math. 20, 177-190, 1933.Cipra, B. "If You Can't See It, Don't Believe It...." Science 259, 26-27, 1993.Cipra, B. What's Happening in the Mathematical Sciences, Vol. 1. Providence, RI: Amer. Math. Soc., pp. 21-25, 1993.Grinsztajn, M. "A 63-Dimensional Counterexample to Borsuk's Conjecture." 2026. https://github.com/maaxgrin/borsuk-63-counterexample.Grünbaum, B. "Borsuk's Problem and Related Questions." In Convexity: Proceedings of the Seventh Symposium in Pure Mathematics of the American Mathematical Society, Held at the University of Washington, Seattle, June 13-15, 1961 (Ed. V. Klee). Providence, RI: Amer. Math. Soc., pp. 271-284, 1963.Hinrichs, A. and Richter, C. "New Sets with Large Borsuk Numbers." Disc. Math. 270, 137-147, 2003.Jenrich, T. and Brouwer, A. E. "A 64-Dimensional Counterexample to Borsuk's Conjecture." Elec. J. Combin. 21, P4.29, 2014. https://doi.org/10.37236/4069.Kahn, J. and Kalai, J. K. G. "A Counterexample to Borsuk's Conjecture." Bull. Amer. Math. Soc. 29, 60-62, 1993.Kalai, G. "Progress on Borsuk's Problem and Answer to TYI 60." Combinatorics and More. Oct. 21, 2026. https://gilkalai.wordpress.com/2026/10/21/progress-on-borsuks-problem-and-answer-to-tyi-60/.Lyusternik, L. and Schnirel'mann, L. Topological Methods in Variational Problems. Moscow, Russia, 1930.Lyusternik, L. and Schnirel'mann, L. "Topological Methods in Variational Problems and Their Application to the Differential Geometry of Surfaces." Uspehi Matem. Nauk (N.S.) 2, 166-217, 1947.MathVet. "Which of OpenAI's 372 Mathematical Results Are Machine-Checked as Stated?" 2026. https://math.vet/.Nilli, A. "On Borsuk's Problem." Jerusalem Combinatorics '93. Papers from the International Conference on Combinatorics Held in Jerusalem, May 9-17, 1993 (Ed. H. Barcelo and G. Kalai). Providence, RI: Amer. Math. Soc., pp. 209-210, 1994.OpenAI. "A Nine-Dimensional Counterexample to Borsuk's Covering Assertion." Sep. 23, 2026. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf.Tolmachev, A. "Novel Upper Bounds for Borsuk Numbers from Contact Points and Spherical Partitions." 12 Nov 2026. https://arxiv.org/abs/2611.16775v1.Voronov, V. "A Counterexample to Gale's Conjecture in R^3." 20 Oct 2026. https://arxiv.org/abs/2610.28828.

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Borsuk's Conjecture

Cite this as:

Weisstein, Eric W. "Borsuk's Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BorsuksConjecture.html

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