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Weak square bracket relations for $$P_{\kappa } (\lambda )$$. (English) Zbl 1177.03047
Let $$\kappa$$ and $$\lambda$$ be infinite cardinals with $$\lambda \geq \kappa$$. For $$X,Y \subseteq P(P_{\kappa}(\lambda))$$ and a cardinal $$\rho$$, $$X \rightarrow [Y]^2_{\rho}$$ has the following meaning: Given $$F : P_{\kappa}(\lambda) \times P_{\kappa}(\lambda) \rightarrow \rho$$ and $$A \in X$$, one can find $$B \in Y \cap P(A)$$ and $$\xi < \rho$$ so that $$F(a,b) \neq \xi$$ for all $$a,b \in B$$ with $$a \subset b$$.
Todorčević has shown that that $$\{ P_{\kappa}(\lambda) \} \rightarrow [I_{\kappa,\lambda}^+]_{\lambda}^2$$ does not hold.
This led to the following weaker partition relation:
For $$X,X \subseteq P(P_{\lambda}(\lambda))$$ and $$\rho$$ a cardinal, let $$X \overset{w}{\longrightarrow} [Y]_{\rho}^2$$ mean that given $$F : P_{\kappa}(\lambda) \times P_{\kappa}(\lambda) \rightarrow \rho$$ and $$A \in X$$, one can find $$B \in Y \cap P(A)$$ and $$\xi < \rho$$ such that $$\{ b \in B : F(a,b) = \xi\} \in I_{\kappa,\lambda}$$ for every $$a \in B$$.
The author studies this partition relation. His main result asserts that if $$\kappa$$ is an uncountable strongly compact cardinal and $$\mathfrak{d}_{\kappa} \leq \lambda^{<\kappa}$$, then $$I_{\kappa,\lambda}^+ \overset{w}{\longrightarrow} [I_{\kappa,\lambda}^+]_{\lambda^{<\kappa}}^2$$ does not hold.

MSC:
 300 Partition relations 3e+55 Large cardinals
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References:
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