PIMS-UVic Distinguished Lecture: Karen Gunderson

For any \(r\geq 2\), an \(r\)-uniform hypergraph \(\mathcal{H}\), and integer \(n\), the Turán 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 Turán numbers of graphs are well-understood and exact Turán 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 Turán numbers for other small hypergraphs.