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Claim. The answer to Problem 605 is yes, with a stronger bound than the one that first settled it. Konrad J. Swanepoel and Pavel Valtr, The unit distance problem on spheres, in Towards a Theory of Geometric Graphs (J. Pach, ed.), Contemporary Mathematics 342, American Mathematical Society, 2004, 273–279, write $u_D(n)$ for the largest number of unit distances among $n$ points of the sphere of diameter $D$ in $\mathbb R^3$. Their Theorem 1 states that there is an absolute $c>0$ such that $u_D(n)>c\,n\sqrt{\log n}$ for every $D>1$ and every $n\ge2$. Rescaling the sphere to radius $1$ turns the unit distance into any fixed distance strictly between $0$ and $2$, so $f(n)=c\sqrt{\log n}$ is a function of the kind the problem asks for, and it grows faster than the $c\log^* n$ of Erdős, Hickerson and Pach. The construction places a small cluster $A$ of $t$ points near the equator and takes its images under the rotations of the sphere by the angle sums $\beta(S)$ over subsets $S$ of $A^2$; because these rotations commute, two copies whose index sets differ in one pair contribute a unit distance, which gives at least $\tfrac{t^2}{2}\,2^{t^2}$ unit distances among $t\,2^{t^2}$ points. Theorem 2 of the paper, the same bound for planar sets with no three collinear points and no parallelogram, is not part of this problem. The source card is swanepoel_2004_unit_distance_problem_spheres.

Acceptance. The site's curator, Thomas Bloom, records the theorem as the current lower bound for the problem's quantity, $u_D(n)\gg n\sqrt{\log n}$, beside the solution he credits to Erdős, Hickerson and Pach (problem page accessed); that record is the reviewed evidence. The venue is the one the publisher's record gives, DOI 10.1090/conm/342/06148 (Contemporary Mathematics 342, Towards a Theory of Geometric Graphs, 273–279, 2004), linked above with the first author's publication list. The volume is a proceedings volume rather than a journal, and no evidence that it was refereed is recorded, so no refereed evidence is listed. The proof is unreviewed; acceptance rests on the curator's credit. The site also records the upper bound $u_D(n)\ll n^{4/3}$ for general $D$, so the exact growth of the problem's quantity remains unknown while the question itself is answered.