Let $\mathcal{F}$ be a family of infinite countable sets. I would like to prove the following are equivalent:
1) for each $A\in\mathcal{F}$ exists a finite $F_A\subseteq A$ such that $\{A-F_A | A\in \mathcal{F}\}$ is pairwise disjoint.
2) for each $A_1,A_2\in\mathcal{F}$ such that $A_1\not = A_2$ exists an $n\in\mathbb{N}$ such that $|A_1\cap A_2|<n.$
Are these two really equivalent? or is it only that just one of them implies the other?
They are not equivalent, and though clearly $(1)$ implies $(2)$. To see that they’re not equivalent, suppose that $\mathscr{F}\subseteq\wp(\Bbb N)$ satisfies $(1)$. Let $\mathscr{F}_0=\{A\setminus F_A:A\in\mathscr{F}\}$; this is a pairwise disjoint family of subsets of $\Bbb N$, so it must be countable. Each $A\in\mathscr{F}$ is the union of a member of $\mathscr{F}_0$ and a finite subset of $\Bbb N$, so $\mathscr{F}$ is countable. However, it is well-known that there are families $\mathscr{F}\subseteq\wp(\Bbb N)$ of cardinality $2^\omega=\mathfrak{c}$ that satisfy $(2)$; these are called almost-disjoint families. For constructions of such families see this answer and the answers to this question.
