Combinatorial Aspects of the Splitting Number

Shimon Garti*, Saharon Shelah

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


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 ω) → ( ω) 1,1 2 is consistent with ZFC, even when cf (2 ω1 (hence the weak diamond holds).

Original languageAmerican English
Pages (from-to)709-717
Number of pages9
JournalAnnals of Combinatorics
Issue number4
StatePublished - Dec 2012

Bibliographical note

Funding Information:
∗ Research supported by the United States-Israel Binational Science Foundation. second author.


  • Mathias forcing
  • partition calculus
  • splitting number
  • weak diamond


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