In 1941, Turán showed that the maximum number of edges in an $n$-vertex graph which contains no copy of $K_t$ (the complete $t$-vertex graph) is asymptotically $(1- 1/t)n^2/2$ -- in short, the Turán density of $K_t$ is $1-1/t$. A fundamental theorem due to Erdős, Stone and Simonovits extends this result to arbitrary graphs $H$ in place of $K_t$.
Turán also proposed seemingly simple conjectures on the Turán density of $r$-uniform hypergraphs, which remain unsolved. Our understanding of Turán-type problems for hypergraphs is rather sporadic, so there are around 10 specific hypergraphs whose Turán densities are known. However, these problems drove the development of influential methods, including hypergraph versions of Szemerédi's regularity lemma and the computer-assisted flag-algebra method.
In this talk, I will present the foundations of Turán theory for graphs and hypergaphs, as well as a recent result joint with Letzter and Pokrovskiy.
