KöMaL 2011 A531

Prove that for every positive integer $k$ there is a positive integer $N_k$ (depending only on $k$) such that whenever $\mathcal{C}$ is a set system whose elements are at most $k$-element sets such that every two elements of $\mathcal{C}$ have an element in common, then there exists a set $A$ with at most $N_k$ elements such that every two elements of $\mathcal{C}$ and $A$ have an element in common.