Open hemisphere ($\{x\in S^d:u\cdot x>0\}$)

For a nonzero vector $u$, the open hemisphere of $S^d$ with direction $u$ is $\{x\in S^d:u\cdot x>0\}$. Its boundary is excluded, so opposite open hemispheres are disjoint.

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 $N-d\ge 2r-1$, at least one of each opposite pair of open hemispheres contains $r$ points. Such configurations exist: take distinct real parameters $t_i$ and normalize $(1,t_i,\dots ,t_i^d)$; 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.

For integers $n=2k+d$ with $k\ge 1$ and $d\ge 0$, there are $n$ labelled points of $S^d$ such that every open hemisphere contains at least $k$ of them. The normalized vectors $(-1{)}^i(1,t_i,\dots ,t_i^d)$ for increasing $t_i$ give a signed moment curve construction. A nonzero polynomial of polynomial degree at most $d$ cannot have enough sign changes for fewer than $k$ positive alternating sample values. Zero sample values can be perturbed negatively by a Lagrange interpolation polynomial. Labels allow repeated positions when $d=0$.