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Let
$\kappa $
be a regular uncountable cardinal, and a cardinal greater than or equal to
$\kappa $
. Revisiting a celebrated result of Shelah, we show that if is close to
$\kappa $
and (= the least size of a cofinal subset of ) is greater than , then can be represented (in the sense of pcf theory) as a pseudopower. This can be used to obtain optimal results concerning the splitting problem. For example we show that if and , then no
$\kappa $
-complete ideal on is weakly -saturated.
Let κ be a regular uncountable cardinal and λ be a cardinal greater than κ. We show that if 2<κ ≤ M(κ, λ), then ◇κ,λ holds, where M(κ, λ) equals λℵ0 if cf(λ) ≥ κ, and (λ+)ℵ0 otherwise.
We use the mutually stationary sets of Foreman and Magidor as a tool to establish the validity of the two-cardinal version of the diamond principle in some special cases.
We study the partition relation that is a weakening of the usual partition relation . Our main result asserts that if κ is an uncountable strongly compact cardinal and , then does not hold.
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