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The linear Turán number of the 3-graph $P_5$

Chaoliang Tang, Hehui Wu, Junchi Zhang

Abstract

We prove that for any linear 3-graph on $n$ vertices without a path of length 5, the number of edges is at most $\frac{15}{11}n$, and the equality holds if and only if the graph is the disjoint union of $G_0$, a graph with 11 vertices and 15 edges. Thus, $ex_L(n,P_5)\leq \frac{15}{11}n$, and the equality holds if and only if $11|n$.

The linear Turán number of the 3-graph $P_5$

Abstract

We prove that for any linear 3-graph on vertices without a path of length 5, the number of edges is at most , and the equality holds if and only if the graph is the disjoint union of , a graph with 11 vertices and 15 edges. Thus, , and the equality holds if and only if .
Paper Structure (4 sections, 14 theorems, 1 equation, 23 figures)

This paper contains 4 sections, 14 theorems, 1 equation, 23 figures.

Table of Contents

  1. Introduction
  2. Notations and Sketch of Proof
  3. The Case With Two Disjoint $P_2$
  4. Proof of \ref{['main']}

Key Result

Theorem 1.2

Let $G$ be an $n$ vertex linear 3-graph, containing no $P_5$. Then the number of edges in $G$ is at most $\frac{15}{11}n$, and the equality holds if and only if the graph is the disjoint union of $G_0$, a graph with 11 vertices and 15 edges as shown below.

Figures (23)

  • Figure 1: extremal graph $G_0$
  • Figure 2: Notations for small linear 3-graphs
  • Figure 3: $S_3$ and $P_4$
  • Figure 4: All graphs with $\gamma=2,3$
  • Figure 5: Two cases of connecting outside vertices
  • ...and 18 more figures

Theorems & Definitions (33)

  • Conjecture 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • Definition 3.2
  • Example 3.3
  • Proposition 3.4
  • proof
  • Definition 3.5
  • ...and 23 more