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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:
03E02 Partition relations
03E55 Large cardinals
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