The following question comes from a statement in Joshua Greene's proof of the Kneser conjecture.
He states that, given $n$ and $k$ positive integers, we can find $2n+k$ points on $S^{k+1}$ such that no $k+2$ points of these points lie on a great $k$-sphere.
Why is this true?
A great $k$-sphere is the intersection of $S^{k+1}$ (as a subset of $\mathbb R^{k+2}$) with a $(k+1)$-dimensional hyperplane through the origin. Such a hyperplane is determined by $k+1$ linearly independent points.
Just start out with $k+1$ linearly independent points, and repeatedly add a point that is not on any of the hyperplanes determined by any $k+1$ of the points already present (always possible, since a finite union of hyperplanes will not cover the whole sphere).