Hypergraph Designs

Abstract

Let $H_i$ be the $3$-uniform hypergraph on $4$ vertices with $i$ hyperedges. In this paper, we settle the existence of $H_3$-hypergraph designs of index $\lambda$, obtaining simple $H_3$-hypergraph designs when $\lambda =2$, and providing a new proof of their existence when $\lambda =1$. The existence of simple $H_2$-hypergraph designs of index $\lambda$ is completely settled, as is the spectrum of $H_2$-hypergraph designs of index $\lambda$.