Let $X\neq\emptyset$ be a set. A family ${\cal S}\subseteq {\cal P}(X)$ has property $\mathbf{B}$ if there is $T\subseteq X$ such that for all $S\in{\cal S}$ we have $S\cap T\neq \emptyset$ and $S\not\subseteq T$. Moreover, ${\cal S}$ is said to be linear if $|S_1 \cap S_2| \leq 1$ for all $S_1\neq S_2\in{\cal S}$.

Let $\text{FL}(\omega)$ be the collection of linear families ${\cal S}$ of $\omega$ such that all $S\in{\cal S}$ are finite and have at least $2$ elements. A standard application of Zorn's Lemma shows that every member of $\text{FL}(\omega)$ is contained in a member of $\text{FL}(\omega)$ that is maximal with respect to set inclusion.

Question. Is there a maximal member of $\text{FL}(\omega)$ with property ${\bf B}$?

Let $X\neq\emptyset$ be a set. A family ${\cal S}\subseteq {\cal P}(X)$ has property $\mathbf{B}$ if there is $T\subseteq X$ such that for all $S\in{\cal S}$ we have $S\cap T\neq \emptyset$ and $S\not\subseteq T$. Moreover, ${\cal S}$ is said to be linear if $|S_1 \cap S_2| \leq 1$ for all $S_1\neq S_2\in{\cal S}$.

Let $\text{FL}(\omega)$ be the collection of linear families ${\cal S}$ of $\omega$ such that all $S\in{\cal S}$ are finite and have at least $2$ elements. A standard application of Zorn's Lemma shows that every member of $\text{FL}(\omega)$ is contained in a member of $\text{FL}(\omega)$ that is maximal with respect to set inclusion.

Question. Is there a maximal member of $\text{FL}(\omega)$ which has property ${\bf B}$?