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

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.