Combinatorial Aspects of the Splitting Number

This paper deals with the splitting number s and polarized partition relations. In the first section we define the notion of strong splitting families, and prove that its existence is equivalent to the failure of the polarized relation( ^s _ω) → ( ^s _ω) ^1,1 ₂. We show that the existence of a strong splitting family is consistent with ZFC, and that the strong splitting number equals the splitting number, when it exists. Consequently, we can put some restriction on the possibility that s is singular. In the second section we deal with the polarized relation under the weak diamond, and we prove that the strong polarized relation ^2ω _ω) → ( ^2ω _ω) ^1,1 ₂ is consistent with ZFC, even when cf (2 ^ω=א ₁ (hence the weak diamond holds).