{
  "annotation_id": "01a9b9d5-1525-4d59-b7be-d222947b013d",
  "date_created": "2026-02-17T05:53:20.498000Z",
  "date_modified": "2026-02-17T05:53:20.498000Z",
  "file_hash": "3085dace2a243a1062475c8f50832f07a62a3e022dcc0e9e7151f97a3ff1ff25",
  "private": false,
  "record": {
    "abstract": "We say a hypergraph $\\mathcal{H}$ contains a graph $G$ as trace if there exists a vertex subset $S \\subseteq V(\\mathcal{H})$ such that $|S| = V(G)$ and $\\{e \\cap S \\mid e \\in E(\\mathcal{H})\\}$ contains $G$ as a subgraph.\n  We use $\\mathrm{ex}(n, Tr_r(G))$ to denote the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices not containing $G$ as trace.\n  The study of Tur\\\u0027an numbers for traces was initiated by Mubayi and Zhao~(2017) who studied $\\mathrm{ex}(n, Tr_r(K_{s+1}))$ where $K_{s+1}$ is a clique on $s+1$ vertices and conjectured the exact value of $\\mathrm{ex}(n, Tr_r(K_{s+1}))$.\n  When $r \\le s$, the conjecture was covered by a result of Pikhurko~(2013) who gave the exact value of Tur\\\u0027an numbers for expanded cliques.\n  Then Gerbner and Picollelli~(2023) gave the exact value for book graphs~($K_{1,1,t}$, the complete tripartite graph with two parts of size one and one part of size $t \\ge 2$).\n  We say $G$ is edge-critical if there exists an edge $e \\in E(G)$ such that $\\chi(G - e) \u003c \\chi(G)$ where $\\chi(G)$ is the chromatic number of $G$.\n  The definition of edge-critical was given by Simonovits~(1974), who proved that for an edge-critical graph $G$ with $\\chi(G) = s+1 \\ge 3$, the Tur\\\u0027an graph $T(n,s)$ is the unique extremal graph for $ex(n,G)$ as $n$ is sufficiently large.\n  In this paper, we further generalize the results of Gerbner and Picollelli~(2023) to edge-critical graphs.\n  More precisely, we prove that for an edge-critical graph $G$ with $\\chi(G) = s+1$, when $s \\ge r \\ge 3$ and $n$ is sufficiently large, the $r$-uniform Tur\\\u0027an graph $T_r(n,s)$ is the unique extremal hypergraph.",
    "arxiv_id": "2601.09500",
    "authors": [
      "Yichen Wang",
      "Xin Cheng",
      "Ervin Gy\u0151ri",
      "Yuanpei Wang",
      "Xiamiao Zhao",
      "Junpeng Zhou"
    ],
    "categories": [
      "math.CO"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Forbidding edge-critical graphs as trace in uniform hypergraphs",
    "url": "https://arxiv.org/abs/2601.09500",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "e634d274-830e-4748-9fa5-d6898f5dba8e",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}