Motivation. Yesterday I was sitting idly in the train, contemplating the train network. I noticed that a lot of lines (not all) intersected, and some pairs of lines intersected in quite a few stations. I wondered whether in general it was possible to minimize the intersections in a fictitious train network in which all lines intersected.

Formalization. Let $X\neq \emptyset$ be a set. We say ${\cal C}\subseteq {\cal P}(X)$ is a cover if $\bigcup {\cal C} = X$, and it is intersecting if whenever $A,B\in{\cal C}$ then $A\cap B \neq \emptyset$.

If ${\cal C}, {\cal D} \subseteq {\cal P}(X)$, we say ${\cal C}$ refines ${\cal D}$, or ${\cal C} \preceq {\cal D}$, if for all $C \in {\cal C}$ there is $D\in {\cal D}$ with $C\subseteq D$. Note that $\preceq$ is reflexive and transitive, but not anti-symmetric in general.

We say that an intersecting cover ${\cal C}_0$ is refinement-minimal if whenever ${\cal C}$ is an intersecting cover with ${\cal C} \preceq {\cal C}_0$, then ${\cal C}_0\preceq {\cal C}$.

Question. If $X$ is a non-empty set and ${\cal C} \subseteq {\cal P}(X)$ is an intersecting cover of $X$, does ${\cal C}$ necessarily have a refinement-minimal refinement ${\cal C}_0\preceq {\cal C}$?

Motivation. Yesterday I was sitting idly in the train, contemplating the train network. I noticed that a lot of lines (not all) intersected, and some pairs of lines intersected in quite a few stations. I wondered whether in general it was possible to minimize the intersections in a fictitious train network in which all lines intersected.

Formalization. Let $X\neq \emptyset$ be a set. We say ${\cal C}\subseteq {\cal P}(X)$ is a cover if $\bigcup {\cal C} = X$, and it is intersecting if whenever $A,B\in{\cal C}$ then $A\cap B \neq \emptyset$.

If ${\cal C}, {\cal D} \subseteq {\cal P}(X)$, we say ${\cal C}$ refines ${\cal D}$, or ${\cal C} \preceq {\cal D}$, if for all $C \in {\cal C}$ there is $D\in {\cal D}$ with $C\subseteq D$. Note that $\preceq$ is reflexive and transitive, but not anti-symmetric in general.

We say that an intersecting cover ${\cal C}_0$ is refinement-minimal if whenever ${\cal C}$ is an intersecting cover with ${\cal C} \preceq {\cal C}_0$, then ${\cal C}_0\preceq {\cal C}$.

Question. If $X$ is a non-empty set and ${\cal C} \subseteq {\cal P}(X)$ is an intersecting cover of $X$, is there necessarily an intersecting cover ${\cal C}_0\preceq {\cal C}$ such that ${\cal C}_0$ is refinement-minimal?