While doing my research, I came across yet another problem in $\mathbb Z/n\mathbb Z$ (see my previous question on a related matter here).

Let $n$ be a prime and let $k$ be an integer, $1 \leq k \leq n-1$. Determine a number $t$ such that, whichever distinct non-zero elements $i_1, i_2, \ldots, i_t \in \mathbb Z/n\mathbb Z$ you pick, there would always exist distinct indices $t_1$ and $t_2$, $1 \leq t_1, t_2 \leq t$, and an integer $k_1$, $1 \leq k_1 \leq k$, such that

$$ i_{t_2} \equiv k_1i_{t_1} \pmod{n}. $$

I am especially interested in the case when $n$ is prime and $k \approx n/2$. At first I thought that, in this special setting, $t \leq cn/k$ for some constant $c$, but I guess this is way too optimistic. Any thoughts on the problem would be very much appreciated.