Turán numbers of triangular pyramids

Source abstract

Let $\TP_k$ denote the triangular pyramid with $k$ layers. For a graph family $\mathcal F$, let $\ex(n,\mathcal F)$ be the maximum number of edges in an $n$-vertex graph containing no member of $\mathcal F$ as a subgraph. For a graph $H$, let $\cJ(H)$ be the family of subgraph-minimal graphs among all such graphs $H-I$, where $I\subseteq V(H)$ is an independent set. For every integer $k\ge4$ and all sufficiently large $n$, we prove that \[ \ex(n,\TP_k)=\max_{0\le m\le n}\{m(n-m)+\ex(m,\cJ(\TP_k))\}+O_k(n)=\left\lfloor n^2/4\right\rfloor+Θ_k(n^{4/3}). \] Moreover, for $k=4$, we characterize all extremal graphs and show that \[ \ex(n,\TP_4)=\max_{0\le m\le n}\bigl\{m(n-m)+\ex\big(m,\cJ(\TP_4)\big)\bigr\} \] for sufficiently large $n$.