Call a sequence $1<X_1\le \cdots \le X_m\le n$ block-compatible if there is a pairwise balanced block design $A_1,\dots ,A_m\subseteq \{1,\dots ,n\}$ such that $|A_i|=X_i$ for $1\le i\le m$. (A pairwise block design means that every pair in $\{1,\dots ,n\}$ 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$$\ge \exp (c{n}^{1/2}\log n)$$many block-compatible sequences for $\{1,\dots ,n\}$?