 Extremal hypergraph theory

Extremal hypergraph theory asks how many hyperedges a hypergraph can have under a forbidden-configuration condition. For a fixed $r$-uniform hypergraph $F$, its extremal number ${\mathrm{ex}}_r(n,F)$ is the maximum number of hyperedges in an $n$-vertex $r$-uniform hypergraph containing no copy of $F$.

For $r=2$, this is ordinary extremal graph theory. For $r\ge 3$, pairwise adjacency no longer captures how vertices participate jointly, and even natural small forbidden configurations lead to difficult problems. The normalized limiting edge proportion is the Turán density. Thus hypergraph questions extend graph extremal problems while retaining genuinely higher-order incidence information.