Tur\'an numbers for 3-uniform linear paths of length 3

TL;DR

This paper proves an exact formula for the Turán number of a specific 3-uniform linear path of length 3, confirming a conjecture and linking it to the Turán number of a 3-uniform triangle.

Contribution

It establishes the exact Turán number for the 3-uniform linear path of length 3, confirming a conjecture and connecting it with known results for 3-uniform triangles.

Findings

Exact formula for $ex_3(n; P_3^3)$ for all n.

The Turán number coincides with that of the 3-uniform triangle for large n.

Determination of a conditional Turán number related to $P_3^3$ and $C_3^3$.

Abstract

In this paper we confirm a conjecture of F\"uredi, Jiang, and Seiver, and determine an exact formula for the Tur\'an number $ex_3(n; P_3^3)$ of the 3-uniform linear path $P^3_3$ of length 3, valid for all $n$. It coincides with the analogous formula for the 3-uniform triangle $C^3_3$, obtained earlier by Frankl and F\"uredi for $n\ge 75$ and Cs\'ak\'any and Kahn for all $n$. In view of this coincidence, we also determine a `conditional' Tur\'an number, defined as the maximum number of edges in a $P^3_3$-free 3-uniform hypergraph on $n$ vertices which is \emph{not} $C^3_3$-free.