Countable version of Erdös-Lovasz-Faber conjecture

Let $X$ be an infinite set, and let $(A_n)_ninomega$ be a collection of subsets of $X$ with the following properties:

We consider the following statement:

(EFL$_omega$:) There is $f:Xto omega$ such that for all $ninomega$ the restriction $f|_A_n:A_ntoomega$ is a bijection.

Questions. Is (EFL$_omega$) true? Or does (EFL$_omega$) imply the original Erdös-Faber-Lovasz conjecture?

1 Answer
1

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.

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