@Paul, No, it's not clear (to me, at least). I wrote "if and only if" by mistake when I should have just written "only if". I'll edit and fix it.

Maybe I'm being thick, but is it obvious that $G_{\mathbb N}$ has an infinite independent set?

@Nate: Yes, this was proved by Sierpinski. See at.yorku.ca/p/a/c/a/25.pdf for a proof.

Possibly worth mentioning: if $\mathbb{P}$ is ccc and $|\mathbb{P}| \leq \aleph_1$, then forcing with $\mathbb{P}$ preserves $\diamondsuit$ and CH. This is Exercise IV.7.58 in the newer set theory book by Kunen.

Also, $f_d(k) < R(d+2,\dots,d+2)$ (where $R$ denotes the Ramsey number and there are $k$ entries). This is because you cannot have $d+2$ points that are all mutually the same distance from each other. Suppose you had $R(d+2,\dots,d+2)$ or more points and only $k$ distances represented. Think of these points as the vertices of a complete graph, and think of the distance between two points as the "color" of their edge. The definition of $R$ tells you that you have $d+2$ points all the same distance apart, a contradiction. Thus, for example, $f_3(2)$ is less than $R(5,5) \leq 49$.

Don't know if this helps, but the octahedron is another example showing $f_3(2) \geq 6$.