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Partition calculus and cardinal invariants. (English) Zbl 1322.03033
Recall that \(\binom{\lambda}{\kappa}\rightarrow {\binom{\lambda}{\kappa}}^{1,1}_2\) is an abbreviation of the statement that for every \(c:\lambda\times\kappa\to 2\) there are \(A\subseteq\lambda\), \(B\subseteq\kappa\) such that \(|A|=\lambda\), \(|B|=\kappa\) and \(c\restriction A\times B\) is a constant. In [S. Garti and S. Shelah, Ann. Comb. 16, No. 2, 271–276 (2012; Zbl 1270.03077)], it is shown that if \(\kappa\) is strictly smaller than the splitting number \(\mathfrak{s}\), then \(\binom{\kappa}{\omega}\rightarrow{\binom{\kappa}{\omega}}^{1,1}_2\) holds if and only if \(\text{cf}(\kappa)>\aleph_0\).
In the current paper, the authors show that if \(\kappa\) is a cardinal strictly greater than the reaping number \(\mathfrak{r}\) and \(\kappa\) is of sufficiently large cofinality, then \(\binom{\kappa}{\omega}\rightarrow{\binom{\kappa}{\omega}}^{1,1}_2\) holds. Thus, in the model of A. Blass and S. Shelah [Ann. Pure Appl. Logic 33, 213–243 (1987; Zbl 0634.03047)], where \(\mathfrak{r}=\aleph_1<\mathfrak{s}=\mathfrak{c}=\aleph_2\), the partition relation \(\binom{\Theta}{\omega}\rightarrow{\binom{\Theta}{\omega}}^{1,1}_2\) holds for every \(\Theta\in [\aleph_1, 2^{\aleph_0}]\). The authors also give certain generalizations of the above results to higher cardinals. For example, it is shown that if \(\lambda\) is regular, \(\lambda<\mu<\mathfrak{s}_\lambda\), where \(\mathfrak{s}_\lambda\) denotes the generalized splitting number at \(\lambda\), then \(\binom{\mu}{\lambda}\rightarrow{\binom{\mu}{\lambda}}^{1,1}_2\) if and only if \(\text{cf}(\mu)\neq\lambda\). Also, if \(\lambda\) is regular and \(\mu\) satisfies \(\mathfrak{r}_\lambda<\text{cf}(\mu)\leq\mu\leq 2^\lambda\), where \(\mathfrak{r}_\lambda\) denotes the generalized reaping number at \(\lambda\), then \(\binom{\mu}{\lambda}\rightarrow{\binom{\mu}{\lambda}}^{1,1}_2\).
In addition, for certain singular cardinals \(\mu\) and under certain pcf assumptions, the authors obtain the consistency of a positive polarized relation for many cardinals in the interval \((\mu,2^\mu]\).

MSC:
03E02 Partition relations
03E04 Ordered sets and their cofinalities; pcf theory
03E05 Other combinatorial set theory
03E17 Cardinal characteristics of the continuum
03E35 Consistency and independence results
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References:
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