Turán numbers for 3-uniform linear paths of length 3

Abstract

In this paper we confirm a conjecture of Füredi, Jiang, and Seiver, and determine an exact formula for the Turán number $e{x}_3(n;P_3^3)$ 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 Füredi for $n\ge 75$ and Csákány and Kahn for all $n$. In view of this coincidence, we also determine a `conditional' Turán 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.