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 $k\in {\cal L}$?



Injective choice function for "lines" in an infinite cardinal
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 $f(k\_\beta),\beta<\alpha$.


