Forcing a $\square(\kappa)$-like principle to hold at a weakly compact cardinal

Abstract

Hellsten \cite{MR2026390} proved that when $\kappa$ is $\Pi^1_n$-indescribable, the \emph{$n$-club} subsets of $\kappa$ provide a filter base for the $\Pi^1_n$-indescribability ideal, and hence can also be used to give a characterization of $\Pi^1_n$-indescribable sets which resembles the definition of stationarity: a set $S\subseteq\kappa$ is $\Pi^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteq\kappa$. By replacing clubs with $n$-clubs in the definition of $\Box(\kappa)$, one obtains a $\Box(\kappa)$-like principle $\Box_n(\kappa)$, a version of which was first considered by Brickhill and Welch \cite{BrickhillWelch}. The principle $\Box_n(\kappa)$ is consistent with the $\Pi^1_n$-indescribability of $\kappa$ but inconsistent with the $\Pi^1_{n+1}$-indescribability of $\kappa$. By generalizing the standard forcing to add a $\Box(\kappa)$-sequence, we show that if $\kappa$ is $\kappa^+$-weakly compact and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $\kappa$ remains $\kappa^+$-weakly compact and $\Box_1(\kappa)$ holds. If $\kappa$ is $\Pi^1_2$-indescribable and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $\kappa$ is $\kappa^+$-weakly compact, $\Box_1(\kappa)$ holds and every weakly compact subset of $\kappa$ has a weakly compact proper initial segment. As an application, we prove that, relative to a $\Pi^1_2$-indescribable cardinal, it is consistent that $\kappa$ is $\kappa^+$-weakly compact, every weakly compact subset of $\kappa$ has a weakly compact proper initial segment, and there exist two weakly compact subsets $S^0$ and $S^1$ of $\kappa$ such that there is no $\beta<\kappa$ for which both $S^0\cap\beta$ and $S^1\cap\beta$ are weakly compact.