Injective choice function for "lines" in an infinite cardinal

Question

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

1. $|k| = \lambda$ for all $k\in {\cal L}$ and,
2. 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 $k\in {\cal L}$?

Answer 1

Observe that $|\mathcal L|\leq\lambda$, since mapping $k$ to the pair of its two smallest elements gives an injection $\mathcal L\to\lambda^2$.

Enumerate elements of $\mathcal L$ as $k_\alpha,\alpha<\lambda$. Then we can define by transfinite recursion $f(k_\alpha)$ to be the least element of $k_\alpha$ distinct from $f(k_\beta),\beta<\alpha$.