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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:
 300 Partition relations 30000 Ordered sets and their cofinalities; pcf theory 300000 Other combinatorial set theory 3e+17 Cardinal characteristics of the continuum 3e+35 Consistency and independence results
##### Keywords:
partition calculus; cardinal characteristics; consistency
##### Citations:
Zbl 1270.03077; Zbl 0634.03047
Full Text:
##### References:
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