Isosceles set $i(3)$
The size of the largest set of points in $\mathbb{R}^{3}$ in which every three points form an isosceles triangle.
Value: $8$ Verified lower bound
Verified bounds
Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.
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Lower bound:
$8$
Checked construction
isosceles-set-3-lower-8, submitted by daniel-munro, 2026-11-05
[construction from P. Erdős and L.M. Kelly, Solution to Problem E735: Isosceles n-points, The American Mathematical Monthly 54 (1947) 227-229.] - Upper bound: None yet
Updates from the literature
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1947-04
Lower bound: $8$
P. Erdős and L.M. Kelly, Solution to Problem E735: Isosceles n-points, The American Mathematical Monthly 54 (1947) 227-229.
[via Y.J. Ionin, Isosceles sets, The Electronic Journal of Combinatorics 16 (2009) R141.] -
1962
Upper bound: $8$
H.T. Croft, 9-point and 7-point configurations in 3-space, Proceedings of the London Mathematical Society (3) 12 (1962) 400-424.
[via Y.J. Ionin, Isosceles sets, The Electronic Journal of Combinatorics 16 (2009) R141.]