Given $n$ unit vectors in $\mathbb{R}^n$ s.t. $0 \leq u\cdot v<1$ for all pair of distinct vectors $u,v$. These vectors span a $d$-dimensional subspace s.t. $d< n$. We conjecture that it is possible to partition the $n$ vectors into $d$ groups such that all the vectors within the same group are pairwise non-orthogonal. It trivially holds when $d\in\{1,2,n-1,n-2\}$. However, we have not been able to show for general $d$. Does the conjecture hold for any $d< n$? If yes, how to prove it? Any thoughts/hints would be appreciated. Thanks in advance.



Grouping vectors together
Kahn--Kalai's [counterexample to Borsuk's conjecture](http://arxiv.org/abs/math.MG/9307229) is a collection of vertices of the unit cube.


Imagine that this cube sits in a coordinate hyperplane $x\_1=s$ of $\mathbb R^{n+1}$, so that the origin projects to the center of the cube.
Project this cube centrally to the unit sphere.
For a right choice of $s$, you get a counterexample to your statement.


