{
  "id": "2311.18753",
  "version": "v1",
  "published": "2023-11-30T17:54:13.000Z",
  "updated": "2023-11-30T17:54:13.000Z",
  "title": "A note on extremal constructions for the Erdős--Rademacher problem",
  "authors": [
    "Xizhi Liu",
    "Oleg Pikhurko"
  ],
  "comment": "short note, 13 pages, comments are welcome",
  "categories": [
    "math.CO"
  ],
  "abstract": "For given positive integers $r\\ge 3$, $n$ and $e\\le \\binom{n}{2}$, the famous Erd\\H{o}s--Rademacher problem asks for the minimum number of $r$-cliques in a graph with $n$ vertices and $e$ edges. A conjecture of Lov\\'{a}sz and Simonovits from the 1970s states that, for every $r\\ge 3$, if $n$ is sufficiently large then, for every $e\\le \\binom{n}{2}$, at least one extremal graph can be obtained from a complete partite graph by adding a triangle-free graph into one part. In this note, we explicitly write the minimum number of $r$-cliques predicted by the above conjecture. Also, we describe what we believe to be the set of extremal graphs for any $r\\ge 4$ and all large $n$, amending the previous conjecture of Pikhurko and Razborov.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-30T17:54:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35"
    ],
    "keywords": [
      "extremal constructions",
      "erdős-rademacher problem",
      "minimum number",
      "extremal graph",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}