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For a   $k$  -uniform hypergraph   $F$   let   $ex(n,F)$   be the maximum number of edges of a   $k$  -uniform   $n$  -vertex hypergraph   $H$   which contains no copy of   $F$  . Determining or estimating   $ex(n,F)$   is a classical and central problem in extremal combinatorics. While for   $k=2$   this problem is well understood, due to the work of Turán and of Erdős and Stone, only very little is known for   $k$  -uniform hypergraphs for   $k&gt;2$  . We focus on the case when   $F$   is a   $k$  -uniform hypergraph with three edges on   $k+1$   vertices. Already this very innocent (and maybe somewhat particular looking) problem is still wide open even for   $k=3$  .

On a generalisation of Mantel’s Theorem to Uniformly Dense Hypergraphs