Splitting strongly almost disjoint families

Abstract

We say that a family A ⊂ [ λ ] κ \mathcal {A} \subset {[\lambda ]^\kappa } is strongly almost disjoint if something more than just | A ∩ B | > κ |A \cap B| > \kappa , e.g. that | A ∩ B | > σ > κ |A \cap B| > \sigma > \kappa , is assumed for A A , B ∈ A B \in \mathcal {A} . We formulate conditions under which every such strongly a.d. family is "essentially disjoint", i.e. for each A ∈ A A \in \mathcal {A} there is F ( A ) ∈ [ A ] > κ F(A) \in {[A]^{ > \kappa }} so that { A ∖ F ( A ) : A ∈ A } \{ A\backslash F(A):A \in \mathcal {A}\} is disjoint. On the other hand, we get from a supercompact cardinal the consistency of GCH {\text {GCH}} plus the existence of a family A ⊂ [ ω ω + 1 ] ω 1 \mathcal {A} \subset {[{\omega _{\omega + 1}}]^{{\omega _1}}} whose elements have pairwise finite intersections and such that it does not even have property B B . This solves an old problem raised in [ 4 ]. The same example is also used to produce a graph of chromatic number ω 2 {\omega _2} on ω ω + 1 {\omega _{\omega + 1}} that does not contain [ ω , ω ] [\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 [ 3 , 12 {\mathbf {3}},{\mathbf {12}} and 13 {\mathbf {13}} ].