# Injective choice function for “lines” in an infinite cardinal

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 $$k\in {\cal L}$$?

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 $$f(k_\beta),\beta<\alpha$$.