---

Claim. The answer to Problem 605 is yes. Paul Erdős, Dean Hickerson and János Pach, A problem of Leo Moser about repeated distances on the sphere, Amer. Math. Monthly 96 (1989), no. 7, 569–575, prove two lower bounds on the unit sphere $S^2$ in $\mathbb R^3$. For every $n$ and every $0<\alpha<2$ there is a set of $n$ points on $S^2$ with at least $c\,n\log^* n$ pairs at distance $\alpha$, where $\log^*$ is the iterated logarithm and $c>0$ is absolute; and for the distance $\sqrt2$ there are $n$ points on $S^2$ with at least $c\,n^{4/3}$ pairs at that distance. Since $\log^* n\to\infty$, the first bound gives the function $f(n)=c\log^* n$ the problem asks for, on the sphere of radius $1$, and the second gives the much faster growth $f(n)\gg n^{1/3}$ for one particular distance. Either bound refutes Leo Moser's conjecture that a fixed distance occurs at most linearly often among $n$ points of a sphere. The general distance is reached by an iterated construction that supplies the $\log^*$ factor; the distance $\sqrt2$, the side of an inscribed square of a great circle, is reached by transferring Erdős's point–line incidence construction to the sphere: incident point–line pairs become orthogonal unit vectors, which are $\sqrt2$ apart. The paper also builds $n$ planar points in general position with fewer than $c\,n^{\log3/\log2}$ distinct distances, which is not part of this problem. The source card is erdos_1989_problem_leo_moser_about_repeated_distances.

Acceptance. The paper is refereed: it appeared in the American Mathematical Monthly, volume 96, issue 7 (August–September 1989), 569–575, DOI 10.1080/00029890.1989.11972243, linked above together with the copy on the Rényi Institute's Erdős page. The site's curator, Thomas Bloom, labels the problem PROVED and credits Erdős, Hickerson and Pach with the solution (problem page accessed), which is the reviewed evidence. The proofs are unreviewed; acceptance rests on the refereed publication and the curator's credit. A stronger lower bound for every sphere of diameter above $1$ is the subject of Swanepoel and Valtr's page.