My question above summarizes my question. I have been studying geometric measure theory and encountered the Besicovitch covering lemma. Also an auxiliary result
"Fix an arbitrary set $A \subset X$ and associate a ball $B(a, r(a))$ to each $a \in A$ so that $sup_{a \in A} r(a) < \infty$. The theorem states that there exists a constant, ß = ß (N), depending only on the normed space, such that for some m < ß , one can find m disjoint subsets $A_i \subset A$ with the property that for each set $A¡$, the associated balls are pairwise disjoint and the union $\cup_{1<i<m} \cup_{a \in A_i} B(a, r(a))$ still covers A."
I found online that for R^1, ß = 2 (easy), R^2, ß = 18, and via the paper "On the Besicovitch constant in small dimensions" that there are upper bounds for small n on euclidean space R^n. But I couldn't find any results for exact numbers.
Since this result was published in 1998, I reckon there must have been some progress on these upper bounds but I could not find any results in the literature. Do we known the exact numbers? And up to which n?
P.S. Note that it has been proven that ß(N) is the same as the amount of n-dimensional unit balls that can be packed in a n-dimensional ball of radius 5.