Some Ramsey-Turán type results for hypergraphs
For a fixed k-graph (k-uniform hypergraph) G, the Turan-number ex(n,G) is the maximum number of edges in a k-graph on n vertices that does not contain an isomorphic copy of G. For \(k>2\), this function is very difficult to determine, so related problems are considered. Let \(\pi (G)=\lim_{n\to \infty}ex(n,G)/\left( \begin{matrix} n\\ k\end{matrix} \right).\) If only hypergraphs of order n with independent sets of order o(n) are considered, then corresponding to \(\pi\) (G) is an extremal parameter \(\rho\) (G), which clearly satisfies \(0\leq \rho (G)\leq \pi (G).\) For k- partite k-graphs, Erdős proved \(0=\rho (G)=\pi (G),\) but Erdős and Sos posed the question of the existence of k-graphs G with \(0<\rho (G)<\pi (G).\) The authors prove, but do not exhibit, the existence of infinitely many k-graphs G satisfying \(0<\rho (G)<\pi (G).\) There are other related results concerning the number of copies of special fixed k- graphs in ``dense'' families of k-graphs.
scientific article; zbMATH DE number 4075105