Let $S(\omega)$ denote the collection of "sparse" infinite subsets of $\omega$, that is, $X\subseteq \omega$ is a member of $S(\omega)$ if and only if both $X$ and $\omega\setminus X$ are infinite.
Is there ${\cal S}\subseteq S(\omega)$ such that for all $a\neq b \in \omega$ we have $|\{s\in {\cal S}: \{a,b\} \subseteq s\}| = 1$?
Sure, let $K$ be any countably infinite field and let $P$ be the projective plane (or a higher-dimensional projective space) over $K$. Let $S'$ be the set of lines in $P$ (where a line is regarded as a set of points), and transport the family $S'$ of subsets of $P$ to a family $S$ of subsets of $\omega$ via your favorite enumeration of $P$.
