Generalized Turán problem for Complete Hypergraphs

Write $K_n^{(k)}$ for the complete $k$-graph on $n$ vertices. For $2\le k\le g<r$ integers, let $\pi (n,K_g^{(k)},K_r^{(k)})$ be the maximum density of $K_g^{(k)}$ in $n$ vertex $K_r^{(k)}$-free $k$-graphs. The main contribution of this paper is the upper bound: $\pi (n,K_g^{(k)},K_r^{(k)})\le (1+O\left(n^{-1}\right)){\prod }_{m=k}^g(1-\frac{\left(\genfrac{}{}{0px}{}{m-1}{k-1}\right)}{\left(\genfrac{}{}{0px}{}{r-1}{k-1}\right)})\mathrm{.}$ The graph case ($k=2$) is the first known generalized Turán question, investigated by Erdős. The $k=g$ case is the hypergraph Turán problem where the best known general upper bound is by de Caen. The result proved here matches both bounds asymptotically, while any triple $k,g,r$ with $2<k<g<r$ provides a new upper bound. The proof uses techniques from the theory of flag algebras to derive linear relations between different densities. These relations can be combined with linear algebraic methods. Additionally a simple flag algebraic certificate will be given for ${\lim }_{n\to \mathrm{\infty }}\pi (n,K_4^{(3)},K_5^{(3)})=3\mathrm{/}8$.