{
  "abstract": "We study the forbidden poset analog of the maximal anti-Ramsey problem introduced for graphs by Burr, Erd\\H os, Graham, and Sós. For integers $m\\le 2^n$ and poset $P=(P,\\preceq)$, we introduce $\\mathrm{ar_m}(n,m,P)$ (and $\\mathrm{ar^*_m}(n,m,P)$) to denote the minimum integer $k$ such that there exists a family $\\mathcal{F}\\subseteq 2^{[n]}$ with $|\\mathcal{F}|=m$ and a coloring $\\mathcal{F}\\rightarrow [k]$ with all weak (strong) copies of $P$ being rainbow. As long as there exist $P$-free families of size $m$, these parameters equal 1. It is known that the largest size $La(n,P)$ ($La^*(n,P)$) of weak (strong) $P$-free families has order of magnitude $Θ(\\binom{n}{\\lfloor n/2\\rfloor})$ unless $P$ is the antichain $A_k$ on $k$ elements. In this paper we study $\\mathrm{ar_m}(n,m,P)$ and $\\mathrm{ar^*_m}(n,m,P)$ in two regimes of $m$. We determine the asymptotics of these parameters for all posets $P$ when $m=2^n$. We also consider the case $m=Θ(\\binom{n}{\\lfloor n/2\\rfloor})$. It is shown that for any connected poset $P$ and integer $k$, there exist integers $m_{P,k}$ and $m^*_{P,k}$ such that to color the middle $k$ layers of the Boolean lattice with all weak or strong copies of $P$ being rainbow, one needs $Θ(n^{m_{P,k}})$ or $Θ(n^{m^*_{P,k}})$ colors. For tree posets $T$, one has $m_{T,k}=m^*_{T,k}$. We conjecture that for any tree poset $T$, and positive real $\\varepsilon$, $\\mathrm{ar_m}(n,m,T),\\mathrm{ar^*_m}(n,m,T)=Ω(n^{m_{T,k}})$ holds provided $m\\ge (k-1+\\varepsilon)\\binom{n}{\\lfloor n/2\\rfloor}$. We prove our conjecture on $\\mathrm{ar_m}(n,m,T)$ for an infinite class of tree posets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30354",
  "author_total": 3,
  "authors": [
    {
      "name": "Binlong Li"
    },
    {
      "name": "Balázs Patkós"
    },
    {
      "name": "Changxin Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0hrhekf2s58jc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30354",
  "fulltext_url": null,
  "id": "0hrhekf2s58jc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30354",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7204867178",
  "pdf_url": "https://arxiv.org/pdf/2608.30354",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0hrhekf2s58jc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30354",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30354"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Maximal anti-Ramsey problems for posets",
  "title_text": "Maximal anti-Ramsey problems for posets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/0hrhekf2s58jc",
  "venue": "arXiv",
  "versions": []
}