# Definition A subset $C\subseteq \mathbb{P}$ is called - *$n$-linke*d 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 uncountable $A \subseteq \mathbb{P}$ contains an uncountable $n$-linked subset. %%$A\in\left[\mathbb{P}\right]^{\aleph_{1}}$ %% - $\mathrm{K}_{2}$ is also called the *Knaster property*. Analoguously, $\mathrm{K}_{n}$ is called *Knaster-$n$ property* - $\mathbb{P}$ has *pre-caliber $\omega_{1}$* if every uncountable $A \subseteq \mathbb{P}$ contains an uncountable centered subset. %%$A\in\left[\mathbb{P}\right]^{\aleph_{1}}$ %% # Properties # Relations ![[Pasted image 20230215165610.png]] ## Implied by - [[Sigma-centered and Sigma-linked]] ## Implies - [[Chain conditions|ccc]]