Splitting strongly almost disjoint families

Abstract:

We say that a family $\mathcal {A} \subset {[\lambda ]^\kappa }$ is strongly almost disjoint if something more than just $|A \cap B| < \kappa$, e.g. that $|A \cap B| < \sigma < \kappa$, is assumed for $A$, $B \in \mathcal {A}$. We formulate conditions under which every such strongly a.d. family is "essentially disjoint", i.e. for each $A \in \mathcal {A}$ there is $F(A) \in {[A]^{ < \kappa }}$ so that $\{ A\backslash F(A):A \in \mathcal {A}\}$ is disjoint. On the other hand, we get from a supercompact cardinal the consistency of ${\text {GCH}}$ plus the existence of a family $\mathcal {A} \subset {[{\omega _{\omega + 1}}]^{{\omega _1}}}$ whose elements have pairwise finite intersections and such that it does not even have property $B$. This solves an old problem raised in [4]. The same example is also used to produce a graph of chromatic number ${\omega _2}$ on ${\omega _{\omega + 1}}$ that does not contain $[\omega ,\omega ]$, answering a problem from [5]. We also have applications of our results to "splitting" certain families of closed subsets of a topological space. These improve results from [${\mathbf {3}},{\mathbf {12}}$ and ${\mathbf {13}}$].