Let $X\neq \emptyset$ be a set. We say $E\subseteq {\cal P}(X)$ is a *linear set system* if for all $a\neq b\in X$ there is exactly one $e\in E$ with $\{a,b\}\subseteq e$.


Is there an infinite cardinal $\kappa$ and a linear set system $E\subseteq {\cal P}(\kappa)$ with $\kappa \notin E$ such that for all $x\in \kappa$ we have $|\{e\in E: x\in e\}| < \kappa$?


**EDIT.** Had to exclude cases like $E = \{\kappa\}$ which Noah Schweber made me aware of (thanks!).



Flat linear set systems
Denote $\deg x =|\{e\in E\colon x\in e\}|$.


Take any $e\in E$; choose $x\notin e$. Then $e$ meets any $e'\ni x$ by at most one element, whence $|e|\leq \deg x<\kappa$.


Now, any $x$ is contained in $\deg x<\kappa$ sets of cardinality $<\kappa$ each; moreover (if this is needed), any of them except one has cardinality at most $\deg y<\kappa$ (for an arbitrary $y\neq x$). All this implies that their union is of cardinality $<\kappa$; but their union **is** $\kappa$.


