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2. Games

    1.     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.

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      Problem 2.1.

      [Foreman]
      1. For what values of $\kappa$ and $\gamma$ are these games determined?
      2. How can these games be generalized to, for example, extender sequences? What about strong compactness and supercompactness?
          Known facts:

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

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

      3. 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 $<\gamma$.

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

          Cite this as: AimPL: From ℵ2 to infinity, available at http://aimpl.org/alephtwo.