Turán H-densities for 3-graphs
Given an $r$-graph $H$ on $h$ vertices, and a family $\mathcal{F}$ of forbidden subgraphs, we define $\ex_{H}(n, \mathcal{F})$ to be the maximum number of induced copies of $H$ in an $\mathcal{F}$-free $r$-graph on $n$ vertices. Then the \emph{Turán $H$-density} of $\mathcal{F}$ is the limit \[π_{H}(\mathcal{F})= \lim_{n\rightarrow \infty}\ex_{H}(n, \mathcal{F})/\binom{n}{h}. \] This generalises the notions of \emph{Turán density} (when $H$ is an $r$-edge), and \emph{inducibility} (when $\mathcal{F}$ is empty). Although problems of this kind have received some attention, very few results are known. We use Razborov's semi-definite method to investigate Turán $H$-densities for 3-graphs. In particular, we show that \[π_{K_4^-}(K_4) = 16/27,\] with Turán's construction being optimal. We prove a result in a similar flavour for $K_5$ and make a general conjecture on the value of $π_{K_t^-}(K_t)$. We also establish that \[π_{4.2}(\emptyset)=3/4,\] where 4.2 denotes the 3-graph on 4 vertices with exactly 2 edges. The lower bound in this case comes from a random geometric construction strikingly different from previous known extremal examples in 3-graph theory. We give a number of other results and conjectures for 3-graphs, and in addition consider the inducibility of certain directed graphs. Let $\vec{S}_k$ be the \emph{out-star} on $k$ vertices; i.e{.} the star on $k$ vertices with all $k-1$ edges oriented away from the centre. We show that \[π_{\vec{S}_3}(\emptyset)=2\sqrt3-3,\] with an iterated blow-up construction being extremal. This is related to a conjecture of Mubayi and Rödl on the Turán density of the 3-graph $C_5$. We also determine $π_{\vec{S}_k}(\emptyset)$ when $k=4$, and conjecture its value for general $k$.