LAUSR

A graph is n-existentially closed (n-e.c.) if for any disjoint subsets A, B of vertices with $$\left| {A \cup B} \right| =n$$A∪B=n, there is a vertex $$z \notin A \cup B$$z∉A∪B adjacent to every vertex of A and no vertex of B. For a block design with block set $${\mathcal {B}}$$B, its block intersection graph is the graph whose vertex set is $${\mathcal {B}}$$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 $$\lambda $$λ-fold triple systems with $$\lambda \ge 2$$λ≥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 $$\{1\}$${1}-block intersection graphs are investigated, and the necessary and sufficient conditions for such graphs to be 2-e.c. are established.