Suppose there are 2n vectors $\{m_1,m_2,...,m_n\}$ and $\{\mu_1,\mu_2,...,\mu_n\}$. All vectors are in k-dimensional Euclid space $R^k$. The vectors satisfy: $$m_i \mu_j \leq 0, \quad \forall i<j $$ and $$ m_i \mu_i > 0, \quad \forall i.$$

What I want to prove: $2n \leq k$.

Intuitively I think the bound $2n \leq k$ is correct. It can be achieved by setting $m_i$ and $\mu_i$ to be the canonical base vectors and their adverse.