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Abstract

It is proved that ideal-based forcings with the side condition method of Todorcevic (1984) add no random reals. By applying Judah–Repický's preservation theorem, it is consistent with the covering number of the null ideal being $\aleph _1$ that there are no $S$-spaces, every poset of uniform density $\aleph _1$ adds $\aleph _1$ Cohen reals, there are only five cofinal types of directed posets of size $\aleph _1$, and so on. This extends the previous work of Zapletal (2004).