Turán numbers for a 4-uniform hypergraph

For any $r\geq 2$, an $r$-uniform hypergraph $\mathcal{H}$, and integer $n$, the \emph{Tur\'{a}n number} for $\mathcal{H}$ is the maximum number of hyperedges in any $r$-uniform hypergraph on $n$ vertices containing no copy of $\mathcal{H}$. While the Tur\'{a}n numbers of graphs are well-understood and exact Tur\'{a}n numbers are known for some classes of graphs, few exact results are known for the cases $r \geq 3$. I will present a construction, using quadratic residues, for an infinite family of hypergraphs having no copy of the $4$-uniform hypergraph on $5$ vertices with $3$ hyperedges, with the maximum number of hyperedges subject to this condition. I will also describe a connection between this construction and a `switching' operation on tournaments, with applications to finding new bounds on Tur\'{a}n numbers for other small hypergraphs.