{
  "abstract": "We consider a random subgraph $G_p(n,r,<s)$ of the complete distance graph $G(n,r,<s)$ whose vertices are the $r$-element subsets of the set $\\{1,\\dots,n\\}$ and whose edges join pairs of subsets that intersect in fewer than $s$ elements; each edge survives independently of the others with probability $p$. The independence number of the graph $G(n,r,<s)$ equals $C_{n-s}^{r-s}$ -- this is the classical Erdos-Ko-Rado theorem. We prove that, for $r=r(n)\\to\\infty$, $s=s(n)\\to\\infty$, $s=o(r)$, $r^2=o(n)$ and $p\\ge 16\\,sr^2\\ln(n/r)/n$, with probability tending to 1 the independence number of the random graph $G_p(n,r,<s)$ also equals $C_{n-s}^{r-s}$, i.e., the Erdos-Ko-Rado result is stable under random sparsification of the graph. Thereby, in the range of parameters $s\\to\\infty$, $s=o(r)$, a recent result of Raigorodskii and Karas is strengthened: the lower bound on the probability $p$ that guarantees stability is lowered by a factor of about $r/s$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08897",
  "author_total": 2,
  "authors": [
    {
      "name": "Vsevolod A. Pokhachevskiy"
    },
    {
      "name": "Andrei Raigorodskii"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5vv815vy4jwhm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08897",
  "fulltext_url": null,
  "id": "5vv815vy4jwhm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08897",
  "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.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7212069088",
  "pdf_url": "https://arxiv.org/pdf/2609.08897",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5vv815vy4jwhm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.08897",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08897"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "On the Stability of the Independence Number in Random Distance Graphs",
  "title_text": "On the Stability of the Independence Number in Random Distance Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/5vv815vy4jwhm",
  "venue": "arXiv",
  "versions": []
}