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PreprintCC BY 4.0arXiv

Monochromatic triangles with empty intersection and Kneser Ramsey numbers

  • Igor Araujo
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Abstract

Recently, Heath, McCourt, Parker, Schwieder, and Zerbib initiated the systematic study of the r-Kneser Ramsey number RrKG(s,t) and investigated related Ramsey-type problems. A central motivation for their work comes from a question of Holmsen, Hrusak, and Roldán-Pensado, who asked whether, for n=2k−1 and sufficiently large k, every red/blue edge-coloring of the complete graph on the vertex set V = binom[n]k necessarily contains a monochromatic triangle ABC with A,B,C ∈ V and A ∩ B ∩ C = ∅. Heath, McCourt, Parker, Schwieder, and Zerbib established that this conclusion holds when n ≥ 7k/3 and k≥ 12. We make substantial progress toward the problem of Holmsen, Hrusak, and Roldán-Pensado by proving that the conclusion already holds for every k≥ 2 whenever n≥ 2k+1. In addition, we obtain improved lower bounds for RrKG(s,t) when s and t are fixed and r is sufficiently large.

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Cite

BibTeXRIS
BibTeX
@misc{araujo2026monochromatic,
  title = {{Monochromatic triangles with empty intersection and Kneser Ramsey numbers}},
  author = {Igor Araujo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27157},
  eprint = {2609.27157},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27157},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Monochromatic triangles with empty intersection and Kneser Ramsey numbers
AU  - Igor Araujo
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.27157
UR  - https://arxiv.org/abs/2609.27157
AB  - Recently, Heath, McCourt, Parker, Schwieder, and Zerbib initiated the systematic study of the r-Kneser Ramsey number R_r^KG(s,t) and investigated related Ramsey-type problems. A central motivation for their work comes from a question of Holmsen, Hrusak, and Roldán-Pensado, who asked whether, for n=2k−1 and sufficiently large k, every red/blue edge-coloring of the complete graph on the vertex set V = binom[n]k necessarily contains a monochromatic triangle ABC with A,B,C ∈ V and A ∩ B ∩ C = ∅. Heath, McCourt, Parker, Schwieder, and Zerbib established that this conclusion holds when n ≥ 7k/3 and k≥ 12. We make substantial progress toward the problem of Holmsen, Hrusak, and Roldán-Pensado by proving that the conclusion already holds for every k≥ 2 whenever n≥ 2k+1. In addition, we obtain improved lower bounds for R_r^KG(s,t) when s and t are fixed and r is sufficiently large.
ER  -