Two-coloring the points of a projective plane
Abstract
Let us color the points of a projective plane of odd order n with two colors, say red and blue. Erdős asked the question of determining the maximum number of balanced lines (that have the same number of red and blue points). The answer is n2, and in the corresponding trivial coloring the unbalanced lines go through a point. In this paper, we consider the Galois plane PG(2,q) and determine the number of unbalanced lines for a non-trivial coloring. The problem can also be generalized to (r,b)-colored lines (see Definition 1.1) and in many cases we extend the results to the generalized problem. Our strongest result for Galois planes of prime order p describes all 2-colorings with at most max{3p+1,4p−22} not (r,b)-colored lines (see Theorem 5.13). Finally, in case of the generalized problem, we allow even more not (r,b)-colored lines if r is small and the order is prime (for more details, see Theorem 5.18).
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BibTeX
@misc{blokhuis2026twocoloring,
title = {{Two-coloring the points of a projective plane}},
author = {Aart Blokhuis and Ádám Markó and Tamás Szőnyi and Zsuzsa Weiner},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41357},
eprint = {2610.41357},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41357},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Two-coloring the points of a projective plane
AU - Aart Blokhuis
AU - Ádám Markó
AU - Tamás Szőnyi
AU - Zsuzsa Weiner
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41357
UR - https://arxiv.org/abs/2610.41357
AB - Let us color the points of a projective plane of odd order n with two colors, say red and blue. Erdős asked the question of determining the maximum number of balanced lines (that have the same number of red and blue points). The answer is n^2, and in the corresponding trivial coloring the unbalanced lines go through a point. In this paper, we consider the Galois plane PG(2,q) and determine the number of unbalanced lines for a non-trivial coloring. The problem can also be generalized to (r,b)-colored lines (see Definition 1.1) and in many cases we extend the results to the generalized problem. Our strongest result for Galois planes of prime order p describes all 2-colorings with at most max{3p+1,4p−22} not (r,b)-colored lines (see Theorem 5.13). Finally, in case of the generalized problem, we allow even more not (r,b)-colored lines if r is small and the order is prime (for more details, see Theorem 5.18).
ER -