Two-cardinal diamond and games of uncountable length

Abstract

Let ${\mu, \kappa}$ and ${\lambda}$ be three uncountable cardinals such that ${\mu = {\rm cf} (\mu) < \kappa = {\rm cf} (\kappa) < \lambda.}$ The game ideal ${NG_{\kappa,\lambda}^\mu}$ is a normal ideal on ${P_\kappa (\lambda)}$ defined using games of length ${\mu}$. We show that if ${2^{(\kappa^{\mu})} \leq \lambda}$ and there are no (fairly) large cardinals in an inner model, then the diamond principle ${\diamondsuit_{\kappa,\lambda} [{NG}_{\kappa,\lambda}^\mu]}$ holds. We also show that if ${\diamondsuit_\kappa (S)}$ holds, where S is a stationary subset of ${\kappa}$, then ${\diamondsuit_{\kappa,\lambda} (\{ a\in P_\kappa (\lambda) : a \cap \kappa \in S\})}$ holds.