Hypergraph Designs

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 \(\lambda\).