On the largest element in D(n)-quadruples

Let $n$ be a nonzero integer. A set of nonzero integers $\{a_1,\dots ,a_m\}$ such that $a_i{a}_j+n$ is a perfect square for all $1\le i<j\le m$ is called a $D\left(n\right)$-$m$-tuple. In this paper, we consider the question, for a given integer $n$ which is not a perfect square, how large and how small can be the largest element in a $D\left(n\right)$-quadruple. We construct families of $D\left(n\right)$-quadruples in which the largest element is of order of magnitude ${\left|n\right|}^3$, resp. ${\left|n\right|}^{2∕5}$.