#definition #forcing #forcing_property # Definition A subset $C\subseteq \mathbb{P}$ is called - $n$-linked if any $n$ elements of $C$ are compatible, i.e. have a common extension (perhaps not in $C$ itself). - linked if it is $2$-linked. - centered if it is $n$-lined for every $n$. I.e. every finite subset has a common extension. - Given $n\in\omega$, $\mathbb{P}$ has property $\mathrm{K}_{n}$ if every $A\in\left[\mathbb{P}\right]^{\aleph_{1}}$ contains an uncountable $n$-linked subset. - $\mathrm{K}_{2}$ is also called the **Knaster property**. - $\mathbb{P}$ has pre-caliber $\omega_{1}$ if every $A\in\left[\mathbb{P}\right]^{\aleph_{1}}$ contains an uncountable centered subset. # Properties # Relations ![[Pasted image 20230215165610.png]] ## Implied by - [[Sigma-centered and Sigma-linked]] ## Implies - [[Chain conditions|ccc]]