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Turán's theorem
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==Hypergraphs and the Turán density== There is no analogous of Turán's theorem for <math>k</math>-uniform hypergraphs. In fact, in Turán's original paper{{r|turan}}, he asked for the maximum number of hyperedges an <math>n</math>-vertex <math>3</math>-uniform hypergraph can have without containing the complete <math>3</math>-uniform hypergraph on <math>4</math> vertices, <math>K_4^{(3)}</math>. This maximum number of hyperedges is known as the ''extremal number''. More precisely and more generally, for a hypergraph <math>F</math>, the '''extremal number''' of <math>F</math> for <math>n</math> vertices, ex<math>(n,F)</math>, is the maximum number of hyperedges an <math>n</math>-vertex <math>k</math>-uniform hypergraph can have without containing a copy of <math>F</math>. To obtain a cleaner parameter, the '''Turán density''' of <math>F</math> is defined by the following limit <math display="block"> \pi(F) = \lim_{n\to \infty}\frac{\text{ex}(n,F)}{\binom{n}{k}}.</math> It is easy to see that <math>\text{ex}(n,F)/\tbinom{n}{k}</math> is non increasing sequence, and therefore, the limit above always converges. With this definition, an approximate answer for Turán's question would determine <math>\pi(K_4^{(3)})</math>.
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