Abstract: The Turán number for a fixed r-uniform hypergraph H is the maximum number of hyperedges in any r-uniform hypergraph on n vertices containing no copy of H. I will discuss some exact Turán numbers that are obtained from a `switching operation' on tournaments. Two tournaments on the same vertex set are switching equivalent if one can be obtained from the other by interchanging all edges between two disjoint sets that partition the vertices. In some cases, infinite classes of extremal hypergraphs can be constructed by taking as hyperedges all sub-tournaments in a fixed switching equivalence class. The target equivalence class can either be defined using some special classes of tournaments with connections to design theory or by computer search. I will discuss some uniqueness results for the extremal results arising from algebraically-defined tournaments, and some possible avenues for further research directions.
