Two-cardinal diamond and games of uncountable length
Résumé
Let μ,κ and λ be three uncountable cardinals such that μ=cf(μ)<κ=cf(κ)<λ. The game ideal NGμκ,λ is a normal ideal on Pκ(λ) defined using games of length μ. We show that if 2(κμ)≤λ and there are no (fairly) large cardinals in an inner model, then the diamond principle ♢κ,λ[NGμκ,λ] holds. We also show that if ♢κ(S) holds, where S is a stationary subset of κ, then ♢κ,λ({a∈Pκ(λ):a∩κ∈S}) holds.