# Finding a good transversal basis

A hypergraph $$H=(V,E)$$ consists of an non-empty set $$V$$ and a collection $$E\subseteq {\cal P}(V)\setminus \{\emptyset\}$$ of non-empty subsets of $$V$$. A transversal of $$H$$ is a set $$T\subseteq V$$ such that $$|T\cap e| = 1$$ for all $$e\in E$$.

It is easy to see that transversals need not exist: Take $$V = \{0,1,2\}$$ and let $$E$$ be the collection of $$2$$-element subsets of $$V$$.

A transversal basis is a set $$B\subseteq V$$ such that $$|B\cap e|\leq 1$$ for all $$e\in E$$. Setting $$I_B:=\{e\in E:B\cap e\neq \emptyset\}$$, we say that that a transversal basis $$B$$ is good if for all transversal bases $$B_1$$ with $$I_{B}\subseteq I_{B_1}$$ we have $$I_B=I_{B_1}$$.

Question. Does every hypergraph $$H=(V,E)$$ have an good transversal basis?

Let $$V=\mathbb Z,E=\{\{m\in\mathbb Z:m\geq n\}:n\in\mathbb Z\}$$. It's clear every transversal basis has at most one element, but no basis $$\{k\}$$ is good, since $$\{k+1\}$$ intersects more sets from $$E$$.