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Claim. Let $f(d)$ be the largest size of an isosceles set in $\mathbb{R}^d$, a set in which every three points determine an isosceles triangle. Y. J. Ionin, Isosceles sets, Electron. J. Combin. 16 (2009), no. 1, #R141, determines in Section 5 and in the table of Section 1

$$f(1),\dots,f(8)=3,\ 6,\ 8,\ 11,\ 17,\ 28,\ 30,\ 45,$$

and describes every isosceles set of maximum size for $d\le7$: one set up to similarity in each dimension except $d=4$, where there are two (a regular pentagon and a congruent one in an orthogonal plane with their common center, and the ten edge midpoints of a regular simplex with its center). The abstract states the exact answer for $d\le7$ and that Blokhuis's bound $\binom{d+2}{2}$ is attained for $d=1,2,6$ and $8$. Section 5 combines Blokhuis's decomposition theorem for isosceles sets with more than two distances (the paper's Theorem 2.15, from the Blokhuis thesis carded as Blokhuis 1984), the paper's Lemma 5.1 on strongly regular two-distance sets, and Lisoněk's maximum two-distance sets in dimensions up to eight; it cites Kelly for $d=2$, gives its own short argument for $d=3$ to $7$, and for $d=8$ takes Lisoněk's 45-point two-distance set, which meets Blokhuis's bound. The paper's main subject is the binary Hamming space, where it bounds isosceles subsets; those results are outside Problem 503.

Covers. The instances $d\le8$ of the problem, with the values above. The values for $d=2$ and $d=3$ are Kelly's and Croft's, recorded on Kelly's claim page and Croft's claim page; the values for $d=4$ to $8$ are first determined here, the value $11$ for $d=4$ independently on Kido's claim page. Nothing is claimed about $d\ge9$.

Depends on. Kelly's claim page, cited for the plane. The two-distance values are Lisoněk's (J. Combin. Theory Ser. A 77 (1997), 318–338) and the decomposition theorem is Blokhuis's, both literature the paper cites.

Acceptance. The paper is refereed: The Electronic Journal of Combinatorics 16 (2009), no. 1, Research Paper 141, published 24 November 2009, as its record gives it; the paper thanks the referee. The site labels the problem OPEN and does not cite the paper, so no reviewed evidence is listed.