A Note on Strongly Almost Disjoint Families

Abstract

For a set $M$, let $\left|M\right|$ denote the cardinality of $M$. A family $\mathcal{F}$ is called strongly almost disjoint if there is an $n\in \omega$ such that $|A\cap B|<n$ for any two distinct elements $A$, $B$ of $\mathcal{F}$. It is shown in $\mathsf{ZF}$ (without the axiom of choice) that, for all infinite sets $M$ and all strongly almost disjoint families $\mathcal{F}\subseteq \mathcal{P}\left(M\right)$, $\left|\mathcal{F}\right|<\left|\mathcal{P}\right(M\left)\right|$ and there are no finite-to-one functions from $\mathcal{P}\left(M\right)$ into $\mathcal{F}$, where $\mathcal{P}\left(M\right)$ denotes the power set of $M$.