### Clubs on quasi measurable cardinals

by Kumar and Shelah. [KmSh:1063]

Math Logic Quarterly, 2018

We construct a model satisfying kappa < 2^{aleph_0}
+ clubsuit_{kappa} + kappa is quasi measurable. Here, we call
kappa
quasi measurable if there is an aleph_1-saturated kappa-additive
ideal {I} over kappa . We also show that, in this model, forcing
with {P}(kappa)/ {I} adds one but not
kappa Cohen reals. We introduce a weak club principle and use
it to show that, consistently, for some aleph_{{1}}-saturated
kappa-additive ideal {I} over kappa, forcing with
P(kappa) /I adds one but not kappa random reals.

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