Piercing spanned lines and hyperplanes

Source abstract

Motzkin, Grünbaum, and Erdős and Purdy asked how many noncollinear blue points can coexist with $t$ red points in the real plane if every blue-spanned line contains a red point. We prove that there are at most $2.35t-3$ blue points, even over the complex numbers. For the complex higher-dimensional problem of Campbell, Kroeker, and Lund, we prove $b_k(t)\le1.40k^2t$. Colorings of cyclic Dowling geometries give a matching lower bound in order of magnitude for all $k\ge2$ and $t\ge3$. The planar proof uses frame normalization and a second-moment estimate; the higher-dimensional proof combines the degree bound of Kroeker and Lund with principal truncation.