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Claim. Theorem 7.2.5 of A. Blokhuis, Few-distance sets, CWI Tract 7, Mathematisch Centrum, Amsterdam, 1984 (card), states that an isosceles set $X$ in $\mathbb{R}^d$, one in which every three points form an isosceles triangle, satisfies $|X|\le(d+1)(d+2)/2$, with equality only for a two-distance set or a spherical two-distance set together with its center. A set with no three points at pairwise distinct distances is exactly an isosceles set, so in the notation of Problem 1088

$$f_d(3)\le\frac{(d+1)(d+2)}{2}+1.$$

The two-distance sets of size $\binom{d+1}{2}$ give a matching lower bound of the same order, so $f_d(3)=d^2/2+O(d)$, as the site's remarks record through Problem 503.

Covers. The case $n=3$ of the problem's second question: $f_d(3)$ grows polynomially in $d$, so $f_d(3)=2^{o(d)}$. Erdős [Er75f, p. 104] wrote that he and Straus could not prove this even for $n=3$. Nothing is claimed for $n\ge4$.

Depends on. No page of this wiki.

Acceptance. None. The CWI Tract, Blokhuis's thesis, is not a journal publication, so refereed is not listed; and the site labels the problem OPEN, so its remarks are not reviewed evidence.