Erdős Problem #732

Call a sequence \(1< X_1\leq \cdots \leq X_m\leq n\) block-compatible if there is a pairwise balanced block design \(A_1,\ldots,A_m\subseteq \{1,\ldots,n\}\) such that \(\lvert A_i\rvert=X_i\) for \(1\leq i\leq m\). (A pairwise block design means that every pair in \(\{1,\ldots,n\}\) is contained in exactly one of the \(A_i\).) Are there necessary and sufficient conditions for \((X_i)\) to be block-compatible? Is there some constant \(c>0\) such that for all large \(n\) there are \[\geq \exp(c n^{1/2}\log n)\] many block-compatible sequences for \(\{1,\ldots,n\}\)?