## The Magidor function and diamond.(English)Zbl 1237.03031

Let $$\kappa$$ be a regular uncountable cardinal and $$\lambda$$ be a cardinal $$>\kappa$$. The author shows that (Theorem 4) if $$2^{<\kappa}\leq M(\kappa,\lambda)$$, then $$\diamondsuit_{\kappa, \lambda}$$ holds, where $$M(\kappa,\lambda)$$ is the Magidor function, which is equal to $$\lambda^\omega$$ if $$\text{cf}(\lambda) \geq \kappa$$, and $$(\lambda^+)^\omega$$ otherwise. Here the diamond principle $$\diamondsuit_{\kappa,\lambda}$$ is a generalization of Jensen’s classical diamond, and it asserts that there exists a sequence $$\langle s_a: a \in P_\kappa(\lambda)\rangle$$ with $$s_a \subseteq a$$ such that for any $$X \subseteq \lambda$$, $$\{a : s_a = X\cap a\}$$ is a stationary subset of $$P_\kappa(\lambda)$$.
In some sense, this result is optimal due to the well-known fact that, assuming $$\diamondsuit_{\kappa,\lambda}$$, the least cardinality of any closed unbounded subset of $$P_\kappa(\lambda)$$, $$c(\kappa,\lambda)$$, equals $$\lambda^{<\kappa}$$, and a result of Magidor that, given the nonexistence of $$\omega_1$$-Erdős cardinals in the core model $$K$$, $$c(\kappa,\lambda)=M(\kappa,\lambda)$$. Therefore, assuming $$\diamondsuit_{\kappa,\lambda}$$ and there is no $$\omega_1$$-Erdős cardinal in the core model $$K$$, $$2^{<\kappa}\leq M(\kappa,\lambda)$$.
However, the author also exhibits (relative to a large cardinal) a (Cohen) forcing model in which $$2^{<\kappa}> M(\kappa,\lambda)$$ and $$\diamondsuit_{\kappa,\lambda}$$ holds. In fact, a strictly stronger variation of $$\diamondsuit_{\kappa,\lambda}$$, $$\diamondsuit_{\kappa,\lambda}[NG_{\kappa,\lambda}]$$, is instead discussed throughout the paper. A further variation of the form $$\diamondsuit_{\kappa,\lambda,\lambda}[J]$$ is discussed at the end of the paper.

### MSC:

 300000 Other combinatorial set theory

### Keywords:

$$P_\kappa (\lambda )$$; diamond principle
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### References:

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