{
  "abstract": "Given a set $V$ of $n$ vertices and an integer $k\\ge1$, our goal is to maximize the number of edges in graphs $G_1, G_2, \\ldots, G_k$, defined on $V$, under the constraint that the union of all graphs, thought of as a multi-graph, does not contain a rainbow copy of the path $P_5$ on $5$ vertices, that is, a copy of $P_5$ with each of its four edges belonging to a different $G_i$. We consider two versions of the problem, in which, respectively, $\\sum_i e(G_i)$ and $\\min_i e(G_i)$ is maximized. In the former case, we determine the maximum precisely for all $k\\le n-1$ (and also for $P_4$). In the latter, we obtain an asymptotic value for $k\\in\\{5,6,9\\}$ and formulate a very plausible conjecture for all other values of $k$. We also solve the problem for $k=4$, but under an additional assumption of completeness.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.25079",
  "author_total": 4,
  "authors": [
    {
      "name": "Sylwia Antoniuk"
    },
    {
      "name": "Andrzej Grzesik"
    },
    {
      "name": "Magdalena Prorok"
    },
    {
      "name": "Andrzej Ruciński"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/62aqkjp4dxcwy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.25079",
  "fulltext_url": null,
  "id": "62aqkjp4dxcwy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.25079",
  "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": "W7204467781",
  "pdf_url": "https://arxiv.org/pdf/2608.25079",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/62aqkjp4dxcwy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2608.25079",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.25079"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Rainbow Turán numbers for paths of length four",
  "title_text": "Rainbow Turán numbers for paths of length four",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/62aqkjp4dxcwy",
  "venue": "arXiv",
  "versions": []
}