The exact linear Tur\'an number of the Sail

A hypergraph is linear if any two of its edges intersect in at most one vertex. The Sail (or $3$-fan) $F^3$ is the $3$-uniform linear hypergraph consisting of $3$ edges $f_1,f_2,f_3$ pairwise intersecting in the same vertex $v$ and an additional edge $g$ intersecting all $f_i$ in a vertex different from $v$. The linear Tur'an number $e{x}_{lin}(n,F^3)$ is the maximum number of edges in a $3$-uniform linear hypergraph on $n$ vertices that does not contain a copy of $F^3$. F"{u}redi and Gy'arf'as proved that if $n=3k$, then $ex_{lin}(n, F^3) = k^2$ and the only extremal hypergraphs in this case are transversal designs. They also showed that if $n=3k+2$, then $e{x}_{lin}(n,F^3)=k^2+k$, and the only extremal hypergraphs are truncated designs (which are obtained from a transversal design on $3k+3$ vertices with $3$ groups by removing one vertex and all the hyperedges containing it) along with three other small hypergraphs. However, the case when $n=3k+1$ was left open. In this paper, we solve this remaining case by proving that $ex_{lin}(n, F^3) = k^2+1$if$n = 3k+1$, answering a question of F"{u}redi and Gy'arf'as. We also characterize all the extremal hypergraphs. The difficulty of this case is due to the fact that these extremal examples are rather non-standard. In particular, they are not derived from transversal designs like in the other cases.