{
  "abstract": "We use stationary common-neighborhood properties to study highly connected Ramsey relations and partition hypotheses. For weakly compact $κ$, $\\operatorname{Coll}(ω_1,{<}κ)$ forces $ω_2\\to_{\\mathrm{hc},<5}(ω_2)^2_ω$ and $\\operatorname{PH}_1(ω_2)$. If $κ$ is $T^{κ^+}_{ω_1}$-Ramsey, the same collapse forces that every countable coloring of $[ω_2]^2$ has a stationary set $X\\subseteqω_2$ and a color $i$ such that every finite subset of $X$ has stationarily many color-$i$ common neighbors in $X$. From one weakly compact cardinal, we obtain a model of the ${<}5$-edge relation at $ω_3$ and $\\operatorname{PH}_1(ω_3)$, in which $\\check H^2(ω_3,A_d)\\ne0$ for every nontrivial abelian group $A$. This separates $\\operatorname{PH}_1(ω_3)$ from $\\operatorname{PH}_2(ω_3)$, with the exact consistency strength of one weakly compact cardinal. We also show that $\\operatorname{PH}_1(ω_2\\timesω_5)$ is equiconsistent with two weakly compact cardinals.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25092",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiang Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8nj2w5g93b4ss.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25092",
  "fulltext_url": null,
  "id": "8nj2w5g93b4ss",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25092",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7214070080",
  "pdf_url": "https://arxiv.org/pdf/2609.25092",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8nj2w5g93b4ss.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2609.25092",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25092"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Stationary common-neighborhood properties and partition hypotheses",
  "title_text": "Stationary common-neighborhood properties and partition hypotheses",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/8nj2w5g93b4ss",
  "venue": "arXiv",
  "versions": []
}