PreprintCC BY-NC-ND 4.0arXiv
Piercing spanned lines and hyperplanes
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 bk(t)≤1.40k2t. Colorings of cyclic Dowling geometries give a matching lower bound in order of magnitude for all k≥2 and t≥3. 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.
Related
BibTeX
@misc{lund2026piercing,
title = {{Piercing spanned lines and hyperplanes}},
author = {Ben Lund},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25828},
eprint = {2610.25828},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25828},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Piercing spanned lines and hyperplanes
AU - Ben Lund
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25828
UR - https://arxiv.org/abs/2610.25828
AB - 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)≤1.40k^2t. Colorings of cyclic Dowling geometries give a matching lower bound in order of magnitude for all k≥2 and t≥3. 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.
ER -