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 $\mathcal{F}$ of hypergraphs, the linear Tur'an number ${\mathrm{e}\mathrm{x}}_{\mathrm{l}\mathrm{i}\mathrm{n}}(n,\mathcal{F})$ is the maximum number of edges in a $k$-uniform linear hypergraph $\mathcal{H}$ on $n$ vertices that does not contain any member of $\mathcal{F}$ 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 $${\mathrm{e}\mathrm{x}}_{\mathrm{l}\mathrm{i}\mathrm{n}}(n,F^k)\le n^2\mathrm{/}{k}^2\mathrm{.}$$ Moreover, $\mathrm{\mid }\mathcal{H}\mathrm{\mid }=n^2\mathrm{/}{k}^2$ holds if and only if $n\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}k)$ and $\mathcal{H}$ is a transversal design on $n$ points with $k$ groups. We also study ${\mathrm{e}\mathrm{x}}_{\mathrm{l}\mathrm{i}\mathrm{n}}(n,\mathcal{F})$ where $\mathcal{F}$ is any subset of the three linear triple systems with four triples on at most seven points.