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Claim. Let $f(d)$ be the largest size of an isosceles set in $\mathbb{R}^d$, a set in which any three points have two equal distances among them (Definition 1.1 of the paper, which includes collinear triples). Y. J. Ionin, Isosceles sets, Electron. J. Combin. 16 (2009), no. 1, #R141, determines in the table of Section 1 (p. 2), justified in Section 5 (pp. 19--23),

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

A set of points contains three points with three distinct distances exactly when it is not isosceles, so in the notation of Problem 1088 $f_d(3)=f(d)+1$, and

$$f_1(3),\dots,f_8(3)=4,\ 7,\ 9,\ 12,\ 18,\ 29,\ 31,\ 46.$$

The theorem is recorded, as the answers for $d\le8$ of the isosceles-set question, on Ionin's claim page for Problem 503, which describes the method and the extremal sets.

Covers. The values of $f_d(3)$ for $d\le8$, the instances $n=3$, $d\le8$. The values for $d=2$ and $d=3$ are also on the E735 page and Croft's claim page. Kido (Int. J. Combin. 2010, Article ID 803210) found the value $11$ for $d=4$ independently, so $f_4(3)=12$, as Kido's claim page for Problem 503 records. Nothing is claimed for $d\ge9$ or for $n\ge4$.

Depends on. [[problems/discrete_geometry/E1088/claims/1947_04_01_erdos_kelly|The solution of E735]], which the paper cites 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. Refereed: The Electronic Journal of Combinatorics 16 (2009), no. 1, Research Paper 141, published 24 November 2009, as the journal's page gives it. The paper thanks the referee. The erdosproblems.com remarks for this problem do not cite the paper, and Bloom's catalog labeled the problem OPEN, so no reviewed evidence is listed.