The linear Tur\'an number of the k-fan

A hypergraph is linear if any two edges intersect in at most one vertex. For a fixed $k$-uniform family ${\cal{F}}$ of hypergraphs, the linear Tur\'an number ${\rm ex}_{\rm lin}(n,{\cal{F}})$ is the maximum number of edges in a $k$-uniform linear hypergraph $\mathcal H$ on $n$ vertices that does not contain any member of ${\cal{F}}$ as a subhypergraph. For $k\ge 2$ the $k$-fan $F^k$ is the $k$-uniform linear hypergraph having $k$ edges $f_1,\dots,f_k$ pairwise intersecting in the same vertex $v$ and an additional edge $g$ intersecting all $f_i$ in a vertex different from $v$. We prove the following extension of Mantel's theorem $${\rm ex}_{\rm lin}(n,F^k)\le {n^2 / k^2}.$$ Moreover, $|{\mathcal H}|=n^2/k^2$ holds if and only if $n\equiv 0\pmod k$ and $\mathcal H$ is a transversal design on $n$ points with $k$ groups. We also study ${\rm ex}_{\rm lin}(n,{\cal{F}})$ where $\cal{F}$ is any subset of the three linear triple systems with four triples on at most seven points.