{
  "abstract": "We study the separability of subsets of an infinite group. Given an infinite group $G$ and subsets $A,B\\subset G$, we say that $A$ and $B$ are separated in $G$ if there exists an infinite symmetric subset $X\\subset G$ with $e\\in X$ and $XAX\\cap B=\\varnothing$. Question 17.102 of the Kourovka Notebook raises a natural cardinal question: if $A$ and $B$ are disjoint and $|A|,|B|<|G|$, must they be separated? We first give a negative answer, constructing two essentially different families of counterexamples, based respectively on rigid binary relations and on torsion-free groups whose square sets have cardinality smaller than that of the group. On this basis we introduce the $W$-witness set $W_G(A,B)=\\{g\\in G:\\{g,g^{-1}\\}A\\{g,g^{-1}\\}\\cap B=\\varnothing\\}$, obtain a necessary and a sufficient condition for separability given by its size, and study the critical range. The central result of this paper is a necessary and sufficient characterization of separability: taking the inverse pairs $\\{g,g^{-1}\\}$ as vertices, we construct the conflict graph $Γ_{A,B}$, and $A$ and $B$ are separated if and only if the associated graph contains an infinite independent set; separability is thereby converted into an ordinary graph-theoretic problem. From this we further derive the case of finite subsets, the case of Abelian groups, and several cardinal criteria, and we obtain, via Ramsey's theorem, a structural dichotomy for the non-separable case. In addition, we study the range of cardinalities of counterexample groups and the possible sizes of $W$-witness sets, and prove that no uniform necessary and sufficient criterion can depend only on $|A|$, $|B|$ and structural invariants of the group, independently of the specific position of $A$ and $B$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.00504",
  "author_total": 1,
  "authors": [
    {
      "name": "Suhui Wan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a5wkxb374m3xt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.00504",
  "fulltext_url": null,
  "id": "a5wkxb374m3xt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.00504",
  "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.GR"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7172379330",
  "pdf_url": "https://arxiv.org/pdf/2608.00504",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a5wkxb374m3xt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2608.00504",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.00504"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Separability of Subsets in Infinite Groups",
  "title_text": "Separability of Subsets in Infinite Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/a5wkxb374m3xt",
  "venue": "arXiv",
  "versions": []
}