Erdős Problem #665

A pairwise balanced design for \(\{1,\ldots,n\}\) is a collection of sets \(A_1,\ldots,A_m\subseteq \{1,\ldots,n\}\) such that \(2\leq \lvert A_i\rvert <n\) and every pair of distinct elements \(x,y\in \{1,\ldots,n\}\) is contained in exactly one \(A_i\). Is there a constant \(C>0\) and, for all large \(n\), a pairwise balanced design such that \[\lvert A_i\rvert > n^{1/2}-C\] for all \(1\leq i\leq m\)?