An example of an "unlinked" set of $2k + 3$ points in $2k$-space

Abstract

Take any $d + 3$ points in $\mathbb{R}^d$. It is known that (a) if $d = 2k + 1$, then there are two linked $(k + 1)$-simplices with the vertices at these points; (b) if $d = 2k$, then there are two disjoint $(k + 1)$-tuples of these points such that their convex hulls intersect. The analogue of (b) for $d = 2k + 1$, which is also the analogue of (a) for intersections (instead of linkings), states that there are two disjoint $(k + 1)$- and $(k + 2)$-tuples of these points such that their convex hulls intersect. This analogue is correct by (a).