Kiesler and Tarski characterized weakly compact cardinals as those inaccessible cardinals such that for every $\kappa$-complete subalgebra $\mathcal{B}\subseteq P(\kappa))$ every $\kappa$-complete filter on $\mathcal{B}$ can be extended to a $\kappa$-complete ultrafilter on $\mathcal{B}.$ Welch proposed a variant of Holy-Schlict games where, for a fixed $\gamma$, player I and II take turns, with I playing an increasing sequence of subalgebras $\mathcal{A}_{\mathrm{i}}$ and II playing an increasing sequence of ultrafilters $\mathcal{U}_{\mathrm{i}}$ for $ i<\gamma$. Player II wins if she can continue playing of length $\gamma.$
By Kiesler-Tarski, player II wins the game with $\gamma=\omega$ if and only if $\kappa$ is weakly compact. It is immediate that if $\kappa$ is measurable, then II wins the game of length $2^{\kappa}$. Are these the only cases?
Nielsen and Welch proved that if II has a winning strategy in the game of length $\omega+1$ then there is an inner model with a measurable cardinal. Welch conjectured that if II has a winning strategy in the game of length $\omega+1$ then there is a precipitous ideal on $\kappa$ .
Our first result confirms Welch's conjecture: if II has a winning strategy in the game of length $\omega+1$ then there is a normal, $\kappa$-complete precipitous ideal on $\kappa$ . In fact if $\gamma\leq\kappa$ is regular and II wins the game of length $\gamma$, then there is a normal, $\kappa$-complete ideal on $\kappa$ with a dense tree that is $<-\gamma$-closed.
But is this result vacuous? Our second result is that if you start with a model with sufficient fine structure and a measurable cardinal then there is a forcing extension where:
1. $\kappa$ is inaccessible and there is no $\kappa^{+}$-saturated ideal on $\kappa$,
2. for each regular $\gamma\leq\kappa$, player II has a winning strategy in the game of length $\gamma,$
3. for all regular $\gamma\leq\kappa$ there is a normal fine ideal $\mathcal{I}_{\gamma}$ such that $P(\kappa)/\mathcal{I}\gamma$ has a dense, $<-\gamma$ closed tree.
The proofs of these results use techniques from the proofs of determinacy, lottery forcing, iterated club shooting and new techniques in inner model theory. They leave many problems open and not guaranteed to be difficult.
This is joint work of M Foreman, M. Magidor and M. Zeman.[-]