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).
Bibliographical noteFunding Information:
∗ Research supported by the United States-Israel Binational Science Foundation. second author.
- Mathias forcing
- partition calculus
- splitting number
- weak diamond