Combining the theorems of Turán and de Bruijn-Erd\H os

Fix an integer $s\ge 2$. Let $\mathcal{P}$ be a set of $n$ points and let $\mathcal{L}$ be a set of lines in a linear space such that no line in $\mathcal{L}$ contains more than $(n-1)\mathrm{/}(s-1)$ points of $\mathcal{P}$. Suppose that for every $s$-set $S$ in $\mathcal{P}$, there is a pair of points in $S$ that lies in a line from $\mathcal{L}$. We prove that $\mathrm{\mid }\mathcal{L}\mathrm{\mid }\ge (n-1)\mathrm{/}(s-1)+s-1$ for $n$ large, and this is sharp when $n-1$ is a multiple of $s-1$. This generalizes the de Bruijn-Erd\H os theorem which is the case $s=2$. Our result is proved in the more general setting of linear hypergraphs.