Let $S_a^d$ be the $(d-1)$-dimensional sphere of radius $a$ in $\mathbb{R}^d$. Let $r>0$ be a constant and $R=\nu r$ where $\nu>1$ (some constant). Are there any known upper bounds on the number of disjoint $S_r^d$ that can be completely contained in $S_R^d$?
[Math] Sphere packing in a sphere
discrete geometrymg.metric-geometrypacking-and-covering
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I believe what you are seeking is sometimes known as a spherical packing or a spherical code. Here is the MathWorld article on the topic. Here is Neil Sloane's webpage on the topic, including "putative optimal packings" up to $n=130$ spheres. See also integer sequence A126195 for the "Conjectured values for maximal number of solid spheres of radius 1 that can be rolled all in touch with and on the outside surface of a sphere of radius $n$."
There are bounds, but no "analytical result" of the type for which you might be hoping.
Tangential Addendum. There is an interesting variant of the sphere-packing kissing number that seems little known, and for which the bounds are wide enough to invite further work. It is called Hornich's Problem: What is the fewest number of non-overlapping closed unit balls that can radially hide a unit ball $B$, in the sense that every ray from the center of $B$ intersects at least one of the surrounding balls? In $\mathbb{R}^3$, it is known that at least 30 balls are needed, and 42 suffice. So rather different than the kissing number 12!
Described on p.117 of Research Problems in Discrete Geometry, Brass, Peter, Moser, William O. J., Pach, János, 2005. (Springer book link.)
This does not really answer your questions, but I recently got a few results on the chromatic number of the hyperbolic planes. They are formulated by fixing the curvature and letting the distance vary, and I use the notation $\chi(\mathbb{H}^2,\{d\})$ for the chromatic number of the distance-$d$ graph on the hyperbolic plane with curvature $-1$.
for small $d$, $\chi(\mathbb{H}^2,\{d\})\leq 12$ (this can probably be improved, but maybe not easily to $7$),
for large $d$, $\chi(\mathbb{H}^2,\{d\})\leq \frac{4}{\ln 3} d + O(1)$.
The proofs can be found here: https://arxiv.org/abs/1305.2765, published in Geombinatorics Vol XXIV (3) 2015, pp. 117-134 (but the proof of the linear upper bound has some small issues, corrected with an improved bound in the subsequent work of Parlier and Petit https://arxiv.org/abs/1701.08648). All this is not difficult, and the paper raises more questions than it answers.
My impression is that the monotony of the chromatic number with $d$ seems reasonable, but is in fact a subtle issue; and I would rather bet on a negative answer to question (2) but not too high. All in all, these questions are probably incredibly difficult, because we have only very cumbersome tools to relate the geometry with the distance graph.
For the story: about one year after I read, liked and bookmarked your question, I had forgotten about it but read "Ramsey Theory, Today, and Tomorrow", and realized I could answer some questions asked in it by Johnson and Szlam. In the course of writing a paper from these answers, I investigated the case of the hyperbolic plane. After writing a first version I happened to look at my MO favorites, and saw your question again -- which is therefore cited in the paper (will there soon be a @mathoverflow in standard bibTeX definitions?)
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Although the book "Sphere packings, lattices and groups" by Conway and Sloane deals mostly with infinite packings, it has an extensive bibliography that gives some references for the problem you're interested in too. Here are some entries that looked relevant, although I haven't read them: