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.
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Kahn--Kalai' counterexample to Borsuk's conjecture is a collection of vertices of unit cube. Think about this cube as it siitting in an affine hyperplane 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 right choice of parameter you get a counterexample to your statement. |
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