What is the largest m such that there exist $v_1,\dots,v_m \in \mathbb{R}^n$ such that for all $i$ and $j$, if $1\leq i< j\leq m$, then $v_i \cdot v_j < 0$.

Also, the preview screen is not displaying set braces for LaTeX. Is that just the preview, or does it mean the site wouldn't display them after the question had been posted either? (I formatted this question without the braces in case it's the latter.)