Consider $n$ adversarially chosen points in $\mathbb{R}^{d}$ where $n \gg d$. Let $\mathbf{a}$ be one of the $n$ points. Is there an upper bound on the number of points among the remaining $n-1$ points, that can have $\mathbf{a}$ as the nearest neighbor?

It is easy to see that if $\mathbf{a}$ is the nearest neighbor for both $\mathbf{b}$ and $\mathbf{c}$, then if a triangle is drawn with these three points, the angle subtended at $\mathbf{a}$ is at least $60$ degrees.

ais the point with many neighborsbthen the vectors (b-a) / ||b-a|| form a kissing configuration; and conversely such a configuration aroundagivesaas many neighbors as the kissing number. $\endgroup$ – Noam D. Elkies Mar 31 '17 at 3:11