The linear Tur\'an number of the 3-graph $P_5$

TL;DR

This paper determines the maximum number of edges in large linear 3-graphs without a path of length 5, establishing an exact upper bound and characterizing extremal configurations.

Contribution

It introduces the exact linear Turán number for the 3-graph $P_5$ and characterizes the extremal graphs achieving this bound.

Findings

Maximum edges in $P_5$-free linear 3-graphs is $rac{15}{11}n$.

Extremal graphs are disjoint unions of a specific 11-vertex graph.

The bound is tight when $n$ is divisible by 11.

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 $11|n$.