Existential closure in uniform hypergraphs

Abstract

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 $T\subseteq S$, there is a vertex $x\notin S$ adjacent to each vertex of T and no vertex of $S\setminus T$. 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.