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I am very interested in proofs that become shorter and simpler by going to higher dimension in $\mathbb R^n$, or higher cardinality. By "higher" I mean that the proof is using higher dimension or ...

Consider 6 people $p_i$, $i=1,\dots 6$, standing on a sphere $S^2$. We label the positions of these people by $p_i$ again. Suppose no pair of these points $p_i$ are antipodal. At each point $p_i$ ...