Let $\lambda$ be an infinite cardinal and suppose ${\cal L}$ is a collection of subsets of $\lambda$ such that

- $|k| = \lambda$ for all $k\in {\cal L}$ and,
- if $k_1\neq k_2\in {\cal L}$ then $|k_1\cap k_2|\leq 1$.

Is there an injective function $f:{\cal L}\to \lambda$, such that $f(k)\in k$ for all $k\in {\cal L}$?