The Turán Number of the Triangular Pyramid of $3$-Layers

Abstract

The Turán number of a graph $H$, denoted by $\text{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph that does not have $H$ as a subgraph. Let $T{P}_k$ be the triangular pyramid of $k$-layers. In this paper, we determine that $\text{ex}(n,T{P}_3)=\frac{1}{4}{n}^2+n+o(n)$ and pose a conjecture for $\text{ex}(n,T{P}_4)$.