The Tur\'an Number of the Triangular Pyramid of $3$-Layers

TL;DR

This paper determines the maximum number of edges in large graphs avoiding a specific 3-layer triangular pyramid subgraph, providing an exact asymptotic value for the Turán number of $TP_3$ and proposing a conjecture for $TP_4$.

Contribution

It precisely calculates the Turán number for the 3-layer triangular pyramid and introduces a conjecture for the 4-layer case, advancing extremal graph theory.

Findings

$ ext{ex}(n,TP_3)= rac{1}{4}n^2 + n + o(n)$

Established the asymptotic maximum edges avoiding $TP_3$

Proposed a conjecture for the Turán number of $TP_4$

Abstract

The Tur\'an 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 $TP_k$ be the triangular pyramid of $k$-layers. In this paper, we determine that $\text{ex}(n,TP_3)= \frac{1}{4}n^2+n+o(n)$ and pose a conjecture for $\text{ex}(n,TP_4)$.