Unthreadability with Small Conditions

TL;DR

This paper introduces a new forcing method that adds a specific combinatorial sequence under CH and demonstrates how certain square principles can fail in an extended model assuming a weakly compact cardinal.

Contribution

It presents a novel forcing construction that adds a $oxempty(eth_2,eth_0)$-sequence with countable conditions and shows the failure of related square principles under large cardinal assumptions.

Findings

Introduces a forcing adding a $oxempty(eth_2,eth_0)$-sequence with countable conditions.

Shows the failure of $oxempty(eth_2,<eth_0)$ and $oxempty_{eth_1,eth_0}$ in the extension.

Assumes the consistency of a weakly compact cardinal for the main results.

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.