Largest number of two series vectors with negative dot product

Question

Assume $\{x_{i}\}_{i=1}^{m}$, $\{w_{i}\}_{i=1}^{m}$ are two sets of vectors in $\mathbb{R}^{n}$. And we have that $ x_{i}\cdot w_{j} < 0$ for $i \neq j$ and $x_{i}\cdot w_{i} > 0$ for all $i$. I want to prove that $m \leq n+1$.

There is a related question Largest number of vectors with pairwise negative dot product.

Answer 1

For $n\ge 3$ you may have as many such pairs as you wish. Just choose $x_i$ on the boundary of a strictly convex cone with the vertex at $0$ and use the separation theorem to find $w_i$ (all of that can be made completely explicit, of course).