It is proved to be consistent relative to a measurable cardinal that there is a uniform ultrafilter on the real numbers which is generated by fewer than the maximum possible number of sets. It is also shown to be consistent relative to a supercompact cardinal that there is a uniform ultrafilter on \({\aleph }_{\omega +1}\) which is generated by fewer than \({2}^{{\aleph }_{\omega +1}}\) sets.

中文翻译：

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小红衣主教身上的超滤数量少

事实证明，相对于可测量的基数，实数上有一个统一的超滤器是一致的，它由少于最大可能集数生成。相对于超紧凑基数，它也被证明是一致的，\（{\ aleph} _ {\ omega +1} \）上有一个统一的超滤器，它由少于\（{2} ^ {{\\ aleph } _ {\ omega +1}} \）集。