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 ℵ₁ that there are no S-spaces, every poset of uniform density ℵ₁ adds ℵ₁ Cohen reals, there are only five cofinal types of directed posets of size ℵ₁, and so on. This extends the previous work of Zapletal (2004).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm231-2-3,
author = {Teruyuki Yorioka},
title = {Keeping the covering number of the null ideal small},
journal = {Fundamenta Mathematicae},
volume = {228},
year = {2015},
pages = {139-159},
zbl = {06451637},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm231-2-3}
}
Teruyuki Yorioka. Keeping the covering number of the null ideal small. Fundamenta Mathematicae, Tome 228 (2015) pp. 139-159. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm231-2-3/