Keeping the covering number of the null ideal small

摘要

It is proved that ideal-based forcings with the side condition method of Todorcevic (1984) add no random reals. By applying Judah-Repicy's preservation theorem, it is consistent with the covering number of the null ideal being N-1 that there are no S-spaces, every poset of uniform density N-1 adds N-1 Cohen reals, there are only five cofinal types of directed posets of size i, and so on. This extends the previous work of Zapletal (2004).