Games · Question 1

Question context

Let $\kappa$ be a regular cardinal and $\gamma$ an infinite ordinal. Consider the following two-player game (called the Welch game): Player $I$ plays $\kappa$-algebras $\mathscr{A}_i$, while Player II plays increasing $\kappa$-complete filters $\mathcal{F}_i$ on $\mathscr{A}_i$. The game goes on for $\gamma$ many steps. The first player to break one of the rules loses.

Historical workshop record. Current open/solved status has not been independently reassessed.

Status note in the source

Known facts:

• (Keisler-Tarksi) If $\gamma=\omega$, then II has a winning strategy.

• (Foreman-Magidor-Shelah) If II has a winning strategy when $\gamma=\omega+1$, then there is a precipitous ideal on $\kappa$.

• Suppose GCH holds and $\omega_1<\text{cf}(\gamma)=\gamma<\kappa^+$. If II has a winning strategy, then there exists a $\kappa$-complete ideal on $\kappa$ which has a dense tree $T$ of height $\gamma$ which is closed under descending sequences of length $<\gamma$.

• If $\kappa$ is measurable, then II has a winning strategy in the game of length $2^\kappa$.