{
  "abstract": "The celebrated result of Johansson, Kahn and Vu determined the threshold order for clique factors in random graphs, and subsequent work identified the sharp threshold and the corresponding hitting-time phenomenon. In this paper we study the probability that there is no $K_r$-factor above the threshold and, more generally, the probability that the largest $K_r$-matching covers less than $n-s$ vertices of $G(n,p)$. For every fixed $r\\ge3$, throughout the range $$n^{-2/r}(\\log n)^{1/\\binom r2}\\ll p\\ll n^{-2/(r+1)},\\qquad n-s\\in r\\mathbb Z,\\qquad s=o(n),$$ we prove $$\\mathbb P\\bigl(ϕ_r^s(G(n,p))=0\\bigr)=\\exp\\left(-Θ_r\\!\\left((s+1)\\frac{μ_r(n,p)}n\\right)\\right),$$ where $ϕ_r^s(G)$ is the number of $K_r$-matchings covering exactly $n-s$ vertices and $μ_r(n,p):=\\binom nrp^{\\binom r2}$. The lower bound is given by $s+1$ vertices which lie in no copy of $K_r$. For the upper bound we develop an iterable one-root version of the Johansson--Kahn--Vu method. As a structural consequence, we show that the remainder of $G(n,p)$ outside every maximal $K_r$-matching has an almost-perfect $K_{r-1}$-matching throughout the sparse clique window. Independently, we prove a central limit theorem for the maximum $K_r$-matching number. Combining these inputs and a structural theorem for $r=2$ from our earlier work, we prove a central limit theorem for the chromatic number of very dense random graphs: for every $r\\ge2$ and $n^{-2/r}(\\log n)^{1/\\binom r2}\\ll p\\ll n^{-2/(r+1)},$ $$\\frac{χ(G(n,1-p))-\\mathbb Eχ(G(n,1-p))}{\\sqrt{μ_{r+1}(n,p)}/r}\\xrightarrow{\\mathrm d}\\mathcal N(0,1),\\qquad\\operatorname{Var}\\bigl(χ(G(n,1-p))\\bigr) \\sim\\frac{μ_{r+1}(n,p)}{r^2}.$$ This settles the Surya--Warnke conjecture throughout the interior of every clique window with $r\\ge2$, strengthening its concentration prediction to a Gaussian limit with asymptotically exact variance.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12700",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhifei Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ew0x2wn1fkj08.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12700",
  "fulltext_url": null,
  "id": "ew0x2wn1fkj08",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12700",
  "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": "W7212922458",
  "pdf_url": "https://arxiv.org/pdf/2609.12700",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ew0x2wn1fkj08.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.12700",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12700"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Exponential tails for factors and the chromatic number of random graphs",
  "title_text": "Exponential tails for factors and the chromatic number of random graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/ew0x2wn1fkj08",
  "venue": "arXiv",
  "versions": []
}