Polarized Partition Relations and Almost-Disjoint Functions
James E. Baumgartner · Studies in logic and the foundations of mathematics · 1989
Two functions f and g from ω1 to ω are almost disjoint if [a: f( a)=g(a)] is countable. The nonexistence of a family F of pair-wise almost disjoint functions from ω1 to ω with| F| = א2 is known to be of very high consistency strength. If a family of א2 functions is given from ω1 to ω that are almost disjoint taken at a time, then there is such a family that is pair-wise almost disjoint. The main result of this chapter is consistent with CH + 2א1 ≥ א3. The chapter illustrates a theorem that implies the consistency with 2א1 large of the assertion that there is no family of א3 pair-wise almost disjoint functions from ω1 to ω.