Isosceles sets and two-distance sets in Euclidean space

Abstract

A finite set of points in $\mathbb{R}^d$ is isosceles if every three of its points form an isosceles triangle. Erdős asked in 1946 for the largest size $\mathrm{iso}(d)$ of an isosceles set in $\mathbb{R}^d$; the answer was known for $d\le8$. Let $g(d)$ and $M(d)$ be the largest sizes of a two-distance set and of a spherical two-distance set in $\mathbb{R}^d$. We prove that $\mathrm{iso}(d)=\max\{g(d),M(d)+1,M(d-1)+3\}$ for every $d\ge2$, which reduces the problem to these two classical ones. With the recently determined values of $g(d)$ for $9\le d\le22$ we obtain $\mathrm{iso}(d)=\binom{d+1}{2}+1$ for $9\le d\le21$ and $\mathrm{iso}(22)=276$, and we show that the extremal set is unique: for $d\le21$ it is the set of midpoints of the edges of a regular simplex together with its centroid, and for $d=22$ it is the McLaughlin configuration together with its centre. For $10\le d\le22$ every isosceles set with more than $\binom d2+3$ points is a two-distance set or a spherical two-distance set together with its centre. The proofs start from Blokhuis' decomposition of isosceles sets into mutually orthogonal two-distance sets. Counting leaves one configuration open: a two-distance set with $\binom{d+1}{2}$ points in a hyperplane, the largest number allowed by Blokhuis' bound, together with two points that are mirror images of each other. We exclude it by showing that a two-distance set $T$ attaining Blokhuis' bound in $\mathbb{R}^n$ is unisolvent for quadratic polynomials and that, if $T$ lies on two concentric spheres, then apart from two exceptional cases its two layers are spherical $2$-designs whose sizes are determined by the radii. We also show that the squared distances of such a set are in ratio $(k-1):k$ with $(2k-1)^2\le n+1$ if $n\ge9$. Hence there is none for $9\le n\le23$, so that $g(23)\le299$ and $\mathrm{iso}(23)=\max\{g(23),278\}\le299$.