The Tur\'an number of the triangular pyramid of 4-layers

TL;DR

This paper confirms a conjecture about the maximum number of edges in large graphs that do not contain a specific layered triangular pyramid structure, providing new insights into Turán numbers for complex graph configurations.

Contribution

The paper proves Ghosh et al.'s conjecture that the Turán number for the 4-layer triangular pyramid is asymptotically rac{1}{4}n^2 + Θ(n^{4/3}), advancing understanding of Turán problems for layered graphs.

Findings

Confirmed the conjecture on $ex(n, TP_4)$.

Established the asymptotic form of the Turán number for $TP_4$.

Contributed to the theory of Turán numbers for layered graph structures.

Abstract

The Tur\'{a}n number $ex(n,H)$ of a graph $H$ is the maximum number of edges in any $H$-free graph on $n$ vertices. The triangular pyramid of $k$-layers, denoted by $TP_k$, is a generalization of a triangle. The Tur\'an problems of a triangular pyramid with small layers have been studied widely by Liu (E-JC, 2013), Xiao, Katona, Xiao and Zamora (DAM, 2022), Ghosh, Gy\H{o}ri, Paulos, Xiao and Zamora (DAM, 2022). Moreover, Ghosh et al. conjectured that $ex(n, TP_4)=\frac{1}{4}n^2+\Theta(n^{\frac{4}{3}})$. In this note, we confirm this conjecture.