{
  "abstract": "For an integer $k\\ge 3$, let $w(3,k)$ be the least $n$ such that every red-blue coloring of $[n]$ contains either a nontrivial three-term arithmetic progression in blue or a nontrivial $k$-term arithmetic progression in red. A recent breakthrough of Green proved that $w(3,k)\\ge k^{Ω\\left(\\left(\\frac{\\log k}{\\log\\log k}\\right)^{1/3}\\right)}$ when $k$ is large, and Hunter later improved the bound to $w(3,k)\\ge k^{Ω\\left(\\frac{\\log k}{\\log\\log k}\\right)}$. On the other hand, Green remarked that it is reasonable to believe that $w(3,k)\\le k^{O(\\log k)}$. We prove that $w(3,k)\\ge k^{Ω(\\log k)}$, which perhaps gives some evidence that $k^{Θ(\\log k)}$ is the correct order of magnitude.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31798",
  "author_total": 1,
  "authors": [
    {
      "name": "Haitao Cao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9vsjhd1x2ctw2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31798",
  "fulltext_url": null,
  "id": "9vsjhd1x2ctw2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31798",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214770325",
  "pdf_url": "https://arxiv.org/pdf/2609.31798",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9vsjhd1x2ctw2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.31798",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31798"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "An improved lower bound for the van der Waerden number $w(3,k)$",
  "title_text": "An improved lower bound for the van der Waerden number w(3,k)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/9vsjhd1x2ctw2",
  "venue": "arXiv",
  "versions": []
}