An Improvement on Triple Systems Without Two Types of Configurations

There are four nonisomorphic configurations of triples that can form a triangle in a three-uniform hypergraph, where the configurations $B$ and $D$ on $\{1,2,3,4,5\}$ consist of three triples $125,134,234$ and $123,134,235$, respectively. Denote by ex$(n,D)$ and ex$(n,\mathrm{BD})$ the maximum number of triples in a three-uniform hypergraph on $n$ vertices which does not contain $D$, both $B$ and $D$, respectively. Recently, Frankl et al. used theorem of Gustavsson on sufficiently dense graphs to determine ex$(n,D)$ and ex$(n,\mathrm{BD})$ for all $n\ge n_0$. In this note, we use packings and group divisible designs of block size 4 to remove the condition $n\ge n_0$.