A simple \(P_\lambda\)-point on a regular cardinal \(\kappa\) is a uniform ultrafilter on \(\kappa\) with a mod-bounded decreasing generating sequence of length \(\lambda\). We prove that if there is a simple $P_\lambda$-point ultrafilter over $\kappa>\omega$, then $\lambda=\mathfrak{ d}_\kappa=\mathfrak{ b}_\kappa=\mathfrak{ u}_\kappa=\mathfrak{ r}_\kappa=\mathfrak{ s}_\kappa$. We show that such ultrafilters appear in the models of \cite{ SimonOmer,BROOKETAYLOR201737}. We improve the lower bound for the consistency strength of the existence of a $P_{ \kappa^{ ++}}$-point to a $2$-strong cardinal. Finally, we apply our arguments to obtain non-trivial lower bounds for (1) the statement that the generalized tower number $\mathfrak{ t}_\kappa$ is greater than $\kappa^+$ and $\kappa$ is measurable, (2) the preservation of measurability after the generalized Mathias forcing, and (3) variations of filter games of \cite{ NIELSEN_WELCH_2019,HolySchlicht:HierarchyRamseyLikeCardinals,MagForZem} in the case $2^\kappa>\kappa^+$.
Categories: math.LO