Claims

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1933_01_01_borsuk: Every bounded planar set of diameter one is the union of three sets of smaller diameter, the n = 2 instance of the question; refereed in Fund. Math., and credited by the formal-conjectures statement files.

1955_01_01_eggleston: Every set of diameter one in three-dimensional space is the union of four sets of smaller diameter, the n = 3 instance of the question; refereed, and credited by the site and by the formal-conjectures statement file.

1993_07_01_kahn_kalai: Kahn and Kalai answer the question no: finite sets built from equal cuts of a complete graph need at least (1.2)^sqrt(n) parts for large n, and exact counts give failures at n = 1325 and every n > 2014.

2013_05_12_bondarenko: A two-distance set of 416 points on the unit sphere in 65-dimensional space that cannot be split into 83 parts of smaller diameter, so at least 84 are needed where Borsuk's assertion allows 66; refereed in Discrete Comput. Geom.

2014_11_06_jenrich_brouwer: A two-distance set of 352 points in 64-dimensional space whose smaller-diameter subsets have at most five points, so at least 71 parts are needed where Borsuk's assertion allows 65.

2026_05_27_grinsztajn: A public unpublished note claiming 321 points in 63-dimensional space whose smaller-diameter subsets have at most five points, so at least 65 parts are needed where Borsuk's assertion allows 64.

2026_08_12_ji: An arXiv submission claiming the 321-point, 65-part counterexample in 63-dimensional space, generated by ChatGPT and checked by the submitter, withdrawn two days later because the construction had been posted earlier.

2026_09_23_openai: The rank-one projectors of four-dimensional space, with the Frobenius metric, form a compact set of diameter root two in a nine-dimensional space that no ten sets of smaller diameter cover; kernel-checked in Lean.

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