Existential Closure in Uniform Hypergraphs

TL;DR

This paper extends the concept of existential closure from graphs to uniform hypergraphs, establishing conditions for their existence, analyzing random hypergraphs, and providing combinatorial design-based constructions.

Contribution

It introduces the notion of $n$-existential closure for hypergraphs, proves asymptotic properties of random hypergraphs, and offers new combinatorial constructions for such hypergraphs.

Findings

Random uniform hypergraphs are asymptotically $n$-e.c.

Necessary conditions for $n$-e.c. hypergraphs are identified.

Constructions using combinatorial designs generate infinite examples.

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 $x\not\in S$ 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.