For any set $X$ and any cardinal $\kappa$, let $[X]^\kappa$ denote the subsets of $X$ having cardinality $\kappa$.

A *linear hypergraph* is a hypergraph such that for all $e\neq e_1 \in E$ we have $|e\cap e_1|\leq 1$. For any positive integer $n$ let $[n] = \{1,\ldots,n\}$. We say that $([n^2], E)$ is a *square hypergraph* if it is linear and every element of $E$ has $n$ elements.

For any integer $n>1$ let $m(n)$ denote the maximum cardinality of $E$ where $([n^2], E)$ is a square hypergraph. For instance we have $m(2) = 6$. What is the value of $$\lim (\sup)_{n\geq 2}\frac{m(n)}{n^2}\;?$$ (Note: I added $(\sup)$ in case the limit doesn't exist.)