Chain condition for forcing ($\kappa$-cc)

A forcing order has the $\kappa$-chain condition when every antichain has cardinality below $\kappa$. It preserves cardinals and cofinalities at least $\kappa$.

If forcing has the $\kappa$-chain condition for a regular uncountable $\kappa$, every forced club subset of $\kappa$ contains a ground-model club below the forcing condition. Bound the possible witnesses above each ordinal using antichains, then take closure points of the bound function.

Suppose $\mathbb{P}$ has the $\kappa$-chain condition and $\dot{f}$ is forced to be a function from an ordinal $\mu$ to an ordinal $\lambda$. For each $\xi <\mu$, a maximal antichain in a forcing order deciding $\dot{f}(\xi )$ has size below $\kappa$. Hence the ground model has a set $B_{\xi }\subseteq \lambda$ of size below $\kappa$ containing every possible value of $\dot{f}(\xi )$.

If $\kappa$ is a regular cardinal and $\mathbb{P}$ has the $\kappa$-chain condition, then forcing with $\mathbb{P}$ preserves every cardinal number and cofinality at least $\kappa$. Indeed, a proposed surjection $f:\mu \to \lambda$ with $\mu <\lambda$ has its range contained in ${\bigcup }_{\xi <\mu }{B}_{\xi }$ from the possible-values lemma for chain-condition forcing. The regularity of $\kappa$ and infinite cardinal arithmetic make this union have cardinality below the ground-model cardinal $\lambda$, a contradiction.

A forcing order has the countable chain condition when every antichain in a forcing order is countable. Such forcing preserves cardinals and cofinalities at least ${\aleph }_1$.

Every uncountable subset of a Knaster forcing has an uncountable pairwise compatible subset. The property implies the countable chain condition for forcing. Finite Cohen forcing has this property by the delta-system lemma and agreement on the finite root. A Knaster forcing times any CCC forcing is CCC

If $T$ is a normal splitting Suslin tree and $P$ is Knaster, then $P\times T$ is CCC, so $P$ forces that the ground tree is still CCC Countable levels and splitting persist. A new cofinal branch would give an uncountable antichain by splitting, so none is added. This permits adding many Cohen reals while retaining a Suslin tree, and distinguishes Knaster preservation from an unjustified preservation claim for every CCC forcing.