Let $3 \leq k < n \in \mathbb{N}$. By $[n]^k$ we denote the collection of the subsets of $n = \{0,\ldots,n-1\}$ that have size $k$. We say that a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(n,E)$ is $k$-*uniform* if $E\subseteq [n]^k$. Moreover, $H=(n, E)$ is *linear* if $|e\_1\cap e\_2| \leq 1$ for $e\_1\neq e\_2\in E$, and it is *maximal linear* if $E\subseteq E'\subseteq [n]^k$ and $E\neq E'$ imply that $(n, E')$ is no longer linear.


**Question.** Are there integers $k < n$ with $k\geq 3$ and maximal linear hypergraphs $H\_i = (n, E\_i)$ for $i = 1,2$ such that $|E\_1| \neq |E\_2|$? (If yes, it would also be interesting to know how big the difference of the edge sets can become in terms of $n$, but this information is not needed for acceptance of answer.)



Number of edges in $k$-uniform linear hypergraph
For the case $k=3$ we have *partial Steiner triple systems* as a design theory name for $3$-uniform linear hypergraphs. The *spectrum* $S^{(3)}(v)$ consists of the sizes of maximal partial Steiner triple systems taking triples from a $v$-element set. The paper below gives the final steps in determining $S^{(3)}(v)$ for each $v$.
I can access the paper through my university, but I did not find a freely available version.


[The spectrum of maximal partial steiner triple systems](https://doi.org/10.1007/BF01388482)


However, the main result (and other references) can be found in


[Maximal designs and configurations - a survey](https://actamath.savbb.sk/pdf/acta2302.pdf)


looking in Section 9 about partial Steiner triple systems. In particular, you do have multiple sizes in the spectrum. If $v = 12k + r$, then the smallest is around $12k^2$ while the largest is around $24k^2$ (of course with linear and constant terms depending on cases $\pmod{12}$ as well as other caveats).


