= General-position hemisphere count
{title2=$N-d\geq2r-1$}
Choose $N$ points of $S^d$ such that every $d+1$ of their position vectors are <linearly independent>. Any equatorial hyperplane then contains at most $d$ points. If $N-d\geq2r-1$, at least one of each opposite pair of <open hemispheres> contains $r$ points. Such configurations exist: take distinct real parameters $t_i$ and normalize $(1,t_i,\ldots,t_i^d)$; the <Vandermonde determinant> proves the required independence. This weaker one-of-two guarantee already suffices for the topological lower bound in <Lovász theorem on Kneser graphs>.
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