Transversals for strongly almost disjoint families

Abstract:

For a family of sets $A$, and a set $X$, $X$ is said to be a transversal of $A$ if $X\subseteq \bigcup A$ and $|a\cap X|=1$ for each $a\in A$. $X$ is said to be a Bernstein set for $A$ if $\emptyset \not =a\cap X\not =a$ for each $a\in A$. Erdős and Hajnal first studied when an almost disjoint family admits a set such as a transversal or Bernstein set. In this note we introduce the following notion: a family of sets $A$ is said to admit a $\sigma$-transversal if $A$ can be written as $A=\bigcup \{A_n:n\in \omega \}$ such that each $A_n$ admits a transversal. We study the question of when an almost disjoint family admits a $\sigma$-transversal and related questions.