I am looking for the largest $m$ such that there exist $m$ distinct vectors $v_1, v_2, \dots, v_m$ in $\mathbb{F}_3^3$ (i.e. set of 3 element vectors with elements drawn for the field of 3 elements) such that every 3 distinct $v_i, v_j, v_k$ are linearly independent.
I expect there is a largest such $m$. I was easily able to construct a set with $m = 6$ (so we know $m \ge 6$. And obviously $m \le 26$)
Are there any theorems here when the underlying elements are drawn from a field of finite characteristic? (I believe we can show for $\mathbb{R}^n$, the largest such $m$ is infinity)
I can probably brute force this out with a computer, but would be interested in knowing if any such theorems exist.