Let $(M,g)$ be a compact Riemannian manifold. To this Riemannian manifold, we associate a natural number $K(M,g)$ as follows:

$K(M,g)$ is the maximum of all $n\in \mathbb{N}$ such that we have at least $n$ distinct points $x_1,x_2,\ldots,x_n\in M$ such that $d(x_i,x_j)$ is independent of $i\neq j$.

What is this number for the round sphere $S^n$?

Does $K(M,g)$ depends on the Riemannian metric $g$?