Claims

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1947_04_01_erdos_kelly: No seven points of the plane have every triple isosceles, while a regular pentagon with its center does, so f_2(3) = 7; the solution of Monthly problem E735.

1962_01_01_croft: Every nine points of 3-space contain three at pairwise distinct distances, and Kelly's eight-point set does not, so f_3(3) = 9.

1975_01_01_komlos_sulyok_szemeredi: Every set of N integers contains a Sidon subset of size at least c N^{1/2}, so f_1(n) is at most a constant times n^2; with the Erdős–Turán bound this fixes the order n^2 of the one-dimensional case.

1984_01_01_blokhuis: An isosceles set in d-space has at most (d+1)(d+2)/2 points, so f_d(3) is at most (d+1)(d+2)/2 + 1, polynomial in d; this answers the question yes for n = 3. A CWI Tract, not a journal publication.

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