Open on erdosproblems.com (label OPEN; page last edited 18 January 2026). The site records the question as Erdős and Larson's, and Erdős's wider one, for the slowest-growing $h$ such that for all large $n$ some pairwise balanced design has $|A_i|>n^{1/2}-h(n)$ for every block: Erdős and Larson [ErLa82] reach $h(n)\ll n^{1/2-c}$ for some $c>0$, and $h(n)\ll(\log n)^2$ under a Cramér-type bound on prime gaps; Shrikhande and Singhi [ShSi85] embed every large design with blocks of size at least $n^{1/2}-c$ in a projective plane, so the answer is no if every projective plane has prime power order, and, with $H(n)$ the largest gap between consecutive primes up to $n$, the prime power conjecture gives $H(n)\asymp h(n)$. The conditional negative answer is recorded as the accepted conditional claim Shrikhande and Singhi 1985; no unconditional result settles the question.