Hellsten [17] proved that when κ is ${\mathrm{\Pi }}_n^1$-indescribable, the n-club subsets of κ provide a filter base for the ${\mathrm{\Pi }}_n^1$-indescribability ideal, and hence can also be used to give a characterization of ${\mathrm{\Pi }}_n^1$-indescribable sets which resembles the definition of stationarity: a set $S\subseteq \kappa$ is ${\mathrm{\Pi }}_n^1$-indescribable if and only if $S\cap C\ne \varnothing$ for every n-club $C\subseteq \kappa$. By replacing clubs with n-clubs in the definition of $\square (\kappa )$, one obtains a $\square (\kappa )$-like principle ${\square }_n(\kappa )$, a version of which was first considered by Brickhill and Welch [7]. The principle ${\square }_n(\kappa )$ is consistent with the ${\mathrm{\Pi }}_n^1$-indescribability of κ but inconsistent with the ${\mathrm{\Pi }}_{n+1}^1$-indescribability of κ. By generalizing the standard forcing to add a $\square (\kappa )$-sequence, we show that if κ is ${\kappa }^{+}$-weakly compact and GCH holds then there is a cofinality-preserving forcing extension in which κ remains ${\kappa }^{+}$-weakly compact and ${\square }_1(\kappa )$ holds. If κ is ${\mathrm{\Pi }}_2^1$-indescribable and GCH holds then there is a cofinality-preserving forcing extension in which κ is ${\kappa }^{+}$-weakly compact, ${\square }_1(\kappa )$ holds and every weakly compact subset of κ has a weakly compact proper initial segment. As an application, we prove that, relative to a ${\mathrm{\Pi }}_2^1$-indescribable cardinal, it is consistent that κ is ${\kappa }^{+}$-weakly compact, every weakly compact subset of κ has a weakly compact proper initial segment, and there exist two weakly compact subsets $S^0$ and $S^1$ of κ such that there is no $\beta <\kappa$ for which both $S^0\cap \beta$ and $S^1\cap \beta$ are weakly compact.

Hellsten [17]证明当κ为${\mathrm{\Pi }}_{ñ}^{\mathrm{1个}}$难以描述的是，κ的n-club子集为${\mathrm{\Pi }}_{ñ}^{\mathrm{1个}}$-难以刻画的理想状态，因此也可以用来描述 ${\mathrm{\Pi }}_{ñ}^{\mathrm{1个}}$-类似于平稳性定义的可描述集合：一个集合 $\mathrm{小号}\subseteq \kappa$ 是 ${\mathrm{\Pi }}_{ñ}^{\mathrm{1个}}$-仅当且仅当- $\mathrm{小号}\cap C\ne \varnothing$每个n俱乐部$C\subseteq \kappa$。通过在定义中用n -clubs替换clubs$\square （\kappa ）$，一个获得一个 $\square （\kappa ）$类原则 ${\square }_{ñ}（\kappa ）$，Brickhill和Welch [7]首先考虑了其版本。原则${\square }_{ñ}（\kappa ）$ 与 ${\mathrm{\Pi }}_{ñ}^{\mathrm{1个}}$的-indescribability κ但与不一致${\mathrm{\Pi }}_{ñ+\mathrm{1个}}^{\mathrm{1个}}$的-indescribability κ。通过归纳标准强制添加$\square （\kappa ）$序列，我们证明如果κ是${\kappa }^{+}$-弱紧致且GCH保持不变，则存在保留余数的强制扩展，其中κ保留${\kappa }^{+}$-紧凑且 ${\square }_{\mathrm{1个}}（\kappa ）$持有。如果κ是${\mathrm{\Pi }}_{\mathrm{2个}}^{\mathrm{1个}}$-难以描述，并且GCH成立，则存在一个保留余数的强制扩展，其中κ为${\kappa }^{+}$-体积小巧， ${\square }_{\mathrm{1个}}（\kappa ）$保持并且κ的每个弱紧致子集都有一个弱紧致的适当初始段。作为一种应用，我们证明，相对于${\mathrm{\Pi }}_{\mathrm{2个}}^{\mathrm{1个}}$-难以描述的基数，κ是一致的${\kappa }^{+}$-弱紧致，κ的每个弱紧致子集都有一个弱紧致的适当初始段，并且存在两个弱紧致子集${\mathrm{小号}}^0$ 和 ${\mathrm{小号}}^{\mathrm{1个}}$的κ使得没有$\beta <\kappa$ 对于两者 ${\mathrm{小号}}^0\cap \beta$ 和 ${\mathrm{小号}}^{\mathrm{1个}}\cap \beta$ 紧凑。