Unthreadability with Small Conditions

Abstract

We introduce a forcing that adds a $\square ({\aleph }_2,{\aleph }_0)$-sequence with countable conditions under CH. Assuming the consistency of a weakly compact cardinal, we can find a forcing extension by our new poset in which both $\square ({\aleph }_2,<{\aleph }_0)$ and ${\square }_{{\aleph }_1,{\aleph }_0}$ fail in the forcing extension.

Tentative revision; more revisions are still needed, in particular for Section 3