arXiv:2611.05421v1 Announce Type: new Abstract: A finite set of points in $\mathbb{R}^d$ is isosceles if every three of its points form an isosceles triangle. Erd\H{o}s asked in 1946 for the largest size $\mathrm{iso}(d)$ of an isosceles set in $\mathbb{R}^d$. The answer was known for $d\le 8$, the last cases being due to Ionin. We prove that $\mathrm{iso}(d)=\binom{d+1}{2}+1$ for $9\le d\le 21$ and $\mathrm{iso}(22)=276$, and 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, and the number $\binom d2+3$ is attained. The proof combines Blokhuis' decomposition of isosceles sets into mutually orthogonal two-distance sets, for which we give a short self-contained proof (the key step is a special case of Gallai's theorem on edge colourings without rainbow triangles), with the recent determination of the maximum two-distance sets in $\mathbb{R}^d$ for $9\le d\le22$ and with the known maximum sizes of spherical two-distance sets. In dimension nine this gives only $\mathrm{iso}(9)\le47$, and the exact value $46$ requires an additional argument about eight-dimensional two-distance sets with at least $44$ points. We also show that $\mathrm{iso}(23)=\max\{g(23),278\}$, where $g(23)$ is the largest size of a two-distance set in $\mathbb{R}^{23}$, so that $278\le\mathrm{iso}(23)\le575$.
Continue to arXiv Mathematics →You are being forwarded in 6s — stay on this page