  A graph is $n$-existentially closed ($n$-e.c.) if for any disjoint subsets
$A$, $B$ of vertices with $|{A \cup B}|=n$, there is a vertex $z \notin A \cup
B$ adjacent to every vertex of $A$ and no vertex of $B$. For a block design
with block set $\cal B$, its block intersection graph is the graph whose vertex
set is $\cal B$ and two vertices (blocks) are adjacent if they have non-empty
intersection.
  In this paper, we investigate the block intersection graphs of pairwise
balanced designs, and propose a sufficient condition for such graphs to be
$2$-e.c. In particular, we study the $\lambda$-fold triple systems with
$\lambda \ge 2$ and determine for which parameters their block intersection
graphs are $1$- or $2$-e.c. Moreover, for Steiner quadruple systems, the block
intersection graphs and their analogue called $\{1\}$-block intersection graphs
are investigated, and the necessary and sufficient conditions for such graphs
to be $2$-e.c. are established.
