Let $\ S\ $ be a set. A family $\ \{S_i\}_{i\in I}\ $ (where $\ I\ $ is an index set) of subsets of $\ S\ $ are weakly linked (in $\ S\ $) if for every $\ x\in I,\ \exists\ y\in I\setminus \{x\}\ $ such that $\ S_x \cap S_y \neq \emptyset.$
Suppose $\ \{A_i\}_{i\in I}\ \neq \{S\}$ and $\ \{B_j\}_{j\in J}\neq \{S\}\ $ are weakly linked families of (subsets of) $\ S\ $ with $\ \displaystyle\bigcup_{i\in I} A_i = \bigcup_{j\in J} B_j = S.$
I have two questions.
(i) Does there exist $\ i\in I\ $ and $\ K\subset J, K\neq J\ $ such that $\ A_i \subset \displaystyle\bigcup_{j\in K} B_j\ ?$
If not, then:
(ii) Does there exist either $\ i\in I\ $ and $\ K\subset J,\ K\neq J\ $ such that $\ A_i \subset \displaystyle\bigcup_{j\in K} B_j, \ $ or $\ j\in J\ $ and $\ L\subset I,\ L\neq I\ $ such that $\ B_j \subset \displaystyle\bigcup_{i\in L} A_i\ ?$
My thoughts: Note that each family has at least two members (i.e. subsets) in order for the question to not be trivially true due to vacuous truth. Next, the answer to (i) is no. Here is a counter-example: $\ S=\{1,2,3\};\quad \{ A_i \} = \{\ \{1,3\},\ \{1,2,3\}\ \}\quad \{\ B_j\} = \{\ \{ 1,2 \},\ \{ 2,3 \}\ \}.$
Edit: I have also found a counter-example for (ii): $\ \{A_i\} = \{\ \mathbb{Q},\ \left(\mathbb{R}\setminus\mathbb{Q}\right)\cup\{0\}\ \},\ \{B_i\} = \left\{\ \displaystyle\bigcup_{i\ \text{odd}}[i,i+1]\ ,\ \displaystyle\bigcup_{i\ \text{even}}[i,i+1]\right\} $
But I still wonder if there is a counter-example to (ii) in the case where each member of both families are finite sets, or if the set $\ S\ $ is finite or countable.
Here is a counterexample for $(ii)$ with finite $S$: Let $S=\{0, 1, 2, 3\}$ and put $$A=\{S-\{0\}, S-\{1\}\}$$ and $$B=\{S-\{2\}, S-\{3\}\}$$