Let $\mathbb{S}^2$ denote the unit sphere in $\mathbb{R}^3$ with its standard metric, and let $S\subset\mathbb{S}^2$ be some finite set of points. Given $\varepsilon>0$, how big does $S$ need to be to force the minimum non-zero distance between elements of $S$ to be less than $\varepsilon$?
I don't need to know the absolute minimum size of $S$, I just need to know a number $n_\varepsilon\in\mathbb{N}$ such that for any $S\subseteq\mathbb{S}^2$ with $|S|\geq n_\varepsilon$, there are $p,q\in S$ such that $0<d(p,q)<\varepsilon$.
A simple bound can be obtained by noting that if every point is surrounded by a disk of radius $\epsilon$, these cannot wholly cover the sphere as long as
$$S\pi\epsilon^2<4\pi.$$
For small $\epsilon$, the mot compact packing will be hexagonal-like, probably leading to a bound near
$$S=\frac4{\epsilon^2}\frac{2\sqrt3}\pi.$$
https://en.wikipedia.org/wiki/Circle_packing