Existential closure in uniform hypergraphs

For a positive integer n , a graph with at least n vertices is n -existentially closed or simply n -e.c. if for any set of vertices S of size n and any set T ⊆ S , there is a vertex x ∉ S adjacent to each vertex of T and no vertex of S ∖ T . We extend this concept to uniform hypergraphs, find necessary conditions for n -e.c. hypergraphs to exist, and prove that random uniform hypergraphs are asymptotically n -existentially closed. We then provide constructions to generate infinitely many examples of n -e.c. hypergraphs. In particular, these constructions use certain combinatorial designs as ingredients, adding to the ever-growing list of applications of designs.