Definition: Call a set of sets $\mathcal S$ cooperative when for every $\mathcal S'\subseteq\mathcal S$,
- if every two elements in $\mathcal S'$ have nonempty intersection (so $\forall X,Y\in\mathcal S'. X\cap Y\neq\varnothing$)
- then $\mathcal S'$ has nonempty intersection (so $\bigcap \mathcal S'=\varnothing$).
Intuitively, think of $\mathcal S$ as a set of agents (so an agent is an element $X\in\mathcal S$), and imagine that a set of agents can cooperate precisely when its intersection is nonempty. Then $\mathcal S$ is cooperative when a group of agents $\mathcal S'\subseteq \mathcal S$ can cooperate, if and only if its members can cooperate pairwise.
Note that is not required here that $\bigcap\mathcal S\neq\varnothing$, since it need not be the case that every two elements in $\mathcal S$ has nonempty intersection.
Examples:
- $\mathcal S=\bigl\{\{0\},\{1\},\{2\}\bigr\}$ is trivially cooperative, because no distinct elements intersect.
- $\mathcal S=\bigl\{\{0\},\{0,1\},\{0,2\}\bigr\}$ is trivially cooperative, because $\bigcap\mathcal S=\{0\}$.
- $\mathcal S=\bigl\{\{0\},\{0,1\},\{0,2\},\{4\}\bigr\}$ is cooperative (even though $\bigcap\mathcal S=\varnothing$).
- $\mathcal S=\bigl\{ \{0,1\}, \{1,2\}, \{2,0\} \}$ is not cooperative, because if we take $\mathcal S'=\mathcal S$ then every pair of elements in $\mathcal S'$ intersects, but $\bigcap\mathcal S'=\varnothing$.
Has a theory of cooperative sets been explored, and if so where?
Thank you.
(This question is not identical to, but seems related to, Prove that the intersection of all the sets is nonempty. )
