Guessing more sets

Let κ be a regular uncountable cardinal, and λ a cardinal greater than κ with cofinality less than κ. We consider a strengthening of the diamond principle ⋄κ,λ that asserts that any subset of some fixed collection of λ+ elements of Pκ(λ) can be guessed on a stationary set. This new principle, denoted by ⋄κ,λ[λ+], implies that the nonstationary ideal on Pκ(λ) is not 2(λ+)-saturated. We establish that if λ is large enough and there are no inner models with fairly large cardinals, then ⋄κ,λ[λ+] holds. More precisely, it is shown that if 2(κℵ0)≤λ+ and both Shelah's Strong Hypothesis SSH and the Almost Disjoint Sets principle ADSλ hold, then ⋄κ,λ[λ+] holds. The paper also contains ZFC results. Suppose for example that 2κ≤λ+, there is a strong limit cardinal τ with cf(λ)<τ≤κ, and either κ is a successor cardinal greater than ρ+3, where ρ is the largest limit cardinal less than κ, or κ is a limit cardinal and σκ<λ<(σκ)+κ for some cardinal σ≥2. Then ⋄κ,λ[λ+] holds.