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.