What is the largest $m$ such that there exist $v\_1,\dots,v\_m \in \mathbb{R}^n$ such that for all $i$ and $j$, $1\leq i< j\leq m$, we have $v\_i \cdot v\_j < 0$.


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.)



Largest number of vectors with pairwise negative dot product
You can have $m=n+1$. Take the vertices of a regular simplex
with centre at the origin.


You can't have $m=n+2$. There is at least a two-dimensional space
of vectors $(a\_1,\ldots,a\_{n+2})$ such that
$$\sum\_{i=1}^{n+2} a\_i v\_i=0.$$
This gives enough room for manoeuvre
to ensure some $a\_i>0$ and some $a\_j<0$. Thus we get some
nontrivial relation
$$\sum\_{i\in I}a\_i v\_i=\sum\_{j\in J}b\_j v\_j\qquad\qquad(\*)$$
where all the $a\_i>0$ and $b\_j>0$ and $I$ and $J$ are disjoint non-empty
sets of indices. It follows that the dot product of the two sides
of $(\*)$ is negative, but that contradicts it being the square of
the length of
the left side.


