$Fn(\kappa \times \omega , \omega)$ has the countable chain condition.

Question

I am struggling with the following question:
Let $\kappa$ be a regular uncountable cardinal. Show that $Fn(\kappa \times \omega , \omega)$ has the countable chain condition.

Where $Fn(I,\omega)$ is the partial order of all finite partial functions $p: I \rightarrow \omega$ with extension relation superset. (For an infinite index set $I$)

Answer 1

Suppose that $\{p_i\mid i\in I\}$ is an uncountable family of conditions, by the $\Delta$-system lemma, there is an uncountable $J\subseteq I$ such that $\{\operatorname{dom} p_j\mid j\in J\}$ form a $\Delta$-system.

Suppose that the root of the system is $A$ which is a finite subset of $\kappa\times\omega$. There are only countably many functions from $A$ to $\omega$, so there is an uncountable $J'\subseteq J$ such that $\{p_j\mid j\in J'\}$ all agree on their common domain. And of those, any two are compatible.

In fact, the proof shows that this is not only ccc, but in fact Knaster: every uncountable family of conditions has an uncountable subfamily, such that any two are compatible.