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Two-coloring the points of a projective plane — arXiv:2610.41357

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.41357
License (as archived): CC0-1.0
Archived: 2026-11-02T05:00:34.630Z
SHA-256 of the archived text: 0d33b529d24bdb13cf39e37d51b5a13dac62c0e795345537ba3683129807789b
Stable URL of this page: https://metraipolis.com/documento/dc_fa85e702f95a

How to cite

Two-coloring the points of a projective plane — arXiv:2610.41357 — arXiv (en), https://arxiv.org/abs/2610.41357 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_fa85e702f95a (Metraipolis archive dc_fa85e702f95a, sha256 0d33b529d24bdb13cf39e37d51b5a13dac62c0e795345537ba3683129807789b)

BibTeX

@misc{dcfa85e702f95a,
  title = {Two-coloring the points of a projective plane — arXiv:2610.41357},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.41357},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_fa85e702f95a; sha256 0d33b529d24bdb13cf39e37d51b5a13dac62c0e795345537ba3683129807789b},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.41357. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $\mathrm{PG}(2,q)$ 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).

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2610.41357 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-02T04:47:57.678Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).