Let $X$ be an infinite set, and let $(A\_n)\_{n\in\omega}$ be a collection of subsets of $X$ with the following properties:


1. $|A\_m\cap A\_n| \leq 1$ for $m\neq n\in \omega$, and
2. $|A\_n|=\aleph\_0$ for all $n\in \omega$.


We consider the following statement:



> 
> (EFL$\_\omega$:) There is $f:X\to \omega$ such that for all $n\in\omega$ the restriction $f|\_{A\_n}:A\_n\to\omega$ is a bijection.
> 
> 
> 


**Questions.** Is (EFL$\_\omega$) true? Or does (EFL$\_\omega$) imply the original [Erdös-Faber-Lovasz conjecture](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Faber%E2%80%93Lov%C3%A1sz_conjecture)?



Countable version of Erdös-Lovasz-Faber conjecture
If I understand it correctly, it's false. Let $x \notin A\_0 = \{1,2,\dots \}$. Then let $A\_i$ all meet at $x$, and also each meet $A$ at $i$ (add extra elements as necessary; they should be irrelevant). Then $f(x) \neq f(i)$ for any $i$, so $f(x) \notin f(A)$.


This didn't work in the finite case because the sets meeting $A$ cannot exhaust $A$, as the number of such sets is strictly less than the number of elements of $A$. When working over infinite sets, this is no longer true.


