= 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 $\operatorname{ex}_r(n,F)$ 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 $r\ge3$, 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.
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