Kusner's Conjecture

Kusner's conjecture, as recorded by Guy (1983), states that the largest size of an equilateral set in the finite-dimensional normed space satisfies

			(1)
			(2)

Swanepoel (2004) showed that the second assertion fails for some , while Ge et al. (2026) proved that it holds for every when . Xiong (2026) reported that for every there is an for which contains an equilateral set of size . Consequently,

(3)

for .

Xiong (2026) credits GPT-6 Astra with finding the construction, which he then verified and wrote up. As of Sep. 22, 2026, independent specialist review had not been reported.

References

Ge, H.-J.; Xu, Z.; and Zhou, Y. "Kusner's Conjecture: Exact Values and Linear Bounds." 2 Jun 2026. https://arxiv.org/abs/2606.03987.

Guy, R. K. "Unsolved Problems: An Olla-Podrida of Open Problems, Often Oddly Posed." Amer. Math. Monthly 90, 196-200, 1983. https://doi.org/10.2307/2975549.

Swanepoel, K. J. "A Problem of Kusner on Equilateral Sets." Arch. Math. 83, 164-170, 2004. https://doi.org/10.1007/s00013-003-4840-8.

Xiong, N. "Kusner's Conjecture Is False for

." 13 Sep 2026. https://arxiv.org/abs/2609.14794.

Kusner's Conjecture

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