For any set $X$ and cardinal $\kappa$, we denote by
$\ [X]^\kappa :=\ \binom X\kappa\ $ the collection of subsets of $X$ having cardinality $\kappa$.


If $X$ is a set, we call a set system $E\subseteq {\cal P}(X)$ *linear* if for all $a\neq b\in X$ there is exactly one $e\in E$ with $\{a,b\}\subseteq e$. For example, for every set $X$, the set system $[X]^2$ is linear.


For which integers $k\in \omega$ with $k > 2$ is there a linear set system $E\subseteq [\omega]^k$?



$k$-regular linear set systems
I think they exist for all $k \geq 2$. Here is a proof. Fix an enumeration of $[\omega]^2$. For the first step in the process add an arbitrary $k$-set $q\_1=q(p\_1)$ containing the first pair $p\_1$ of the enumeration. At step $i$, if $\ p\_i\subseteq q(p\_j)\ $ for some $\ j<i\ $ then let $\ q(p\_i):=q(p\_j).\ $ Otherwise, select any $(k-2)$-set $\ r\subseteq \omega\setminus p\_i\ $ that is not contained in
$\, \bigcup\_{t=1}^{i-1}\,q(p\_t),\ $ and let


$$ q(p\_i)\ :=\ p\_i\cup r $$


for every $\ i>1.\ $ Then $\ E\ :=\ \{q(p\_i):\ i\in\mathbb N\}\ $
is the solution.


