Claims

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1947_04_01_kelly: No seven points of the plane form an isosceles set, and the only six-point isosceles set is a regular pentagon with its center, so the largest planar isosceles set has 6 points; the solution also gives an 8-point set in space.

1962_01_01_croft: No nine points of three-dimensional space form an isosceles set; with Kelly's eight-point example, the largest isosceles set in $\mathbb{R}^3$ has 8 points.

2009_11_24_ionin: The largest isosceles set in $\mathbb{R}^d$ has 3, 6, 8, 11, 17, 28, 30, 45 points for $d=1,\dots,8$, with every maximum set described for $d\le7$; Blokhuis's bound $\binom{d+2}{2}$ is attained for $d=1,2,6,8$.

2010_01_01_kido: The largest isosceles set in $\mathbb{R}^4$ has 11 points, and there are exactly two 11-point isosceles sets up to isomorphism.

2024_12_06_kovacs: A polynomial-elimination proof that the only six-point isosceles set in the plane is a regular pentagon with its center and that no seven-point set exists, an alternative proof of Kelly's value 6 for $d=2$.

2026_04_22_chojecki: A note proving that the largest isosceles set in $\mathbb{R}^d$ has $\max\{g(d),s(d)+1,s(d-1)+3\}$ points, with $g$ and $s$ the Euclidean and spherical two-distance maxima; it also derives $f(22)=276$ and $f(23)\ge278$.

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