{
  "abstract": "We study a natural extension of Ramsey theory relative to the classes of maximally planar and maximally outerplanar graphs. This can be seen as a continuation of the study of `Planar Ramsey theory', introduced by Axenovich et al. The question we ask is the following: For a fixed family $\\mathcal{K}$ of graphs and a pair of graphs $\\{H,F\\}$, does there exist an integer $r_{\\mathcal{K}} (H, F)$ such that for every graph $G \\in \\mathcal{K}$ with $|G| \\geq r_{\\mathcal{K}}(H, F)$, every red/blue edge-colouring of $G$ admits a red copy of $H$ or a blue copy of $F$? When such an integer exists, we say $\\{H,F\\}$ is unavoidable in $\\mathcal{K}$,, and otherwise $\\{H,F\\}$ is avoidable in $\\mathcal{K}$. Our work focuses on this problem where $\\mathcal{K} = \\mathcal{K}_{\\mathrm{MOP}}$ and $\\mathcal{K} = \\mathcal{K}_{\\mathrm{MP}}$, which denote the families of maximal outerplanar (MOP) graphs and maximal planar (MP) graphs, respectively. This framework generalises the classical Ramsey problem relative to these classes, as the case with $\\mathcal{K} = \\{K_n \\colon n \\geq 2\\}$ corresponds to classical Ramsey. We also study the corresponding Ramsey numbers for MOP and MP, which we denote as $r_{\\mathrm{MOP}}(H, F)$ and $r_{\\mathrm{MP}}(H, F)$. In the case when $\\mathcal{K} = \\mathcal{K}_{\\mathrm{MOP}}$, we completely determine all unavoidable pairs $\\{H, F\\}$ with $|E(F)| \\geq 2$, together with upper bounds and sometimes exact values of $r_{\\mathrm{MOP}}(H, F)$. When $\\mathcal{K} = \\mathcal{K}_{\\mathrm{MP}}$, we completely determine all unavoidable pairs in the diagonal case $\\{H, H\\}$ when $H$ is connected, showing that $H$ must be one of the graphs $P_3$, $P_4$, $P_5$, $K_{1, 3}$ or the fork graph $S_{2,1,1}$. This work opens up further possibilities in the study of Ramsey theory relative to a class, and we offer several open problems in this vein.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05268",
  "author_total": 3,
  "authors": [
    {
      "name": "Adriana Baldacchino"
    },
    {
      "name": "Yair Caro"
    },
    {
      "name": "Xandru Mifsud"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6xqqjay7yr0f3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05268",
  "fulltext_url": null,
  "id": "6xqqjay7yr0f3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05268",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.05268",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6xqqjay7yr0f3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.05268",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05268"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Ramsey properties of maximal (outer)planar graphs",
  "title_text": "Ramsey properties of maximal (outer)planar graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/6xqqjay7yr0f3",
  "venue": "arXiv",
  "versions": []
}