Cardinalities of maximal linear $k$-subsets of $n = \{0,\ldots,n-1\}$

Question

We consider any non-negative integer $n$ as a cardinal, so $0 = \emptyset$, and $n=\{0,\ldots,n-1\}$ for positive $n$. Given $n,k\in \mathbb{N}$, let $[n]^k$ denote the collection of $k$-element subsets of $n$. We say ${\cal L}\subseteq [n]^k$ is linear if $|a\cap b|\leq 1$ whenever $a\neq b\in {\cal L}$, and we call ${\cal L}$ maximal linear if ${\cal M}\subseteq [n]^k$ is no longer linear whenever ${\cal M}\supseteq {\cal L}$ and ${\cal M}\neq {\cal L}$.

Given integers $n\geq k\geq 1$, what is the minimum cardinality of any maximal linear subset of $[n]^k$, and what is the maximum cardinality, in terms of $n,k$?

Answer 1

For the minimum question, this is answered in Theorem 4 of Erdős, Paul; Füredi, Zoltán; Tuza, Zsolt, Saturated (r)-uniform hypergraphs, Discrete Math. 98, No. 2, 95-104 (1991). ZBL0766.05060.

The answer is $\frac{n^2}{k(k-1)^2}+\Theta(n)$ (which is roughly a $\frac{1}{k-1}$ proportion of the maximum possible $\frac{n(n-1)}{k(k-1)}$), and you can get something a bit more precise from their proof if you need.