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The Turán number of the triangular pyramid of 4-layers

Hangdi Chen, Yaojun Chen, Xiutao Zhu

TL;DR

This paper resolves the Turán problem for the triangular pyramid with 4 layers by proving $ex(n,TP_4)=\frac{1}{4}n^2+\Theta(n^{4/3})$. It develops a reduction to TP$_4$-free graphs with large minimum degree and employs a two-part partition analysis, leveraging $ex(n,C_6)$ bounds and bipartite copy results from the Erdős–Simonovits framework. The argument combines a careful structural decomposition with stability-type results to upper-bound the extremal number, matching the known construction's lower bound up to the $\Theta(n^{4/3})$ term. This advances the understanding of Turán-type extremal problems for layered pyramidal graphs and highlights the effectiveness of partition-based and forbidden-subgraph techniques in determining precise asymptotics.

Abstract

The Turán 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 Turán 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, Győri, Paulos, Xiao and Zamora (DAM, 2022). Moreover, Ghosh et al. conjectured that $ex(n, TP_4)=\frac{1}{4}n^2+Θ(n^{\frac{4}{3}})$. In this note, we confirm this conjecture.

The Turán number of the triangular pyramid of 4-layers

TL;DR

This paper resolves the Turán problem for the triangular pyramid with 4 layers by proving . It develops a reduction to TP-free graphs with large minimum degree and employs a two-part partition analysis, leveraging bounds and bipartite copy results from the Erdős–Simonovits framework. The argument combines a careful structural decomposition with stability-type results to upper-bound the extremal number, matching the known construction's lower bound up to the term. This advances the understanding of Turán-type extremal problems for layered pyramidal graphs and highlights the effectiveness of partition-based and forbidden-subgraph techniques in determining precise asymptotics.

Abstract

The Turán number of a graph is the maximum number of edges in any -free graph on vertices. The triangular pyramid of -layers, denoted by , is a generalization of a triangle. The Turán 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, Győri, Paulos, Xiao and Zamora (DAM, 2022). Moreover, Ghosh et al. conjectured that . In this note, we confirm this conjecture.
Paper Structure (3 sections, 9 theorems, 24 equations, 2 figures)

This paper contains 3 sections, 9 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['MainResult']}

Key Result

Theorem 1.1

(Mantel Mantel). For all $n$, $\textup{ex}(n,K_3)=\lfloor\frac{n^2}{4}\rfloor$ and $K_{\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil}$ is the unique extremal graph.

Figures (2)

  • Figure 1: Triangular Pyramids with $2$ and $4$ layers respectively.
  • Figure 2: The graph $F$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 13 more