{
  "annotation_id": "42d938a0-f1d1-44e0-90c9-3441be721c1d",
  "date_created": "2026-02-17T05:53:20.116000Z",
  "date_modified": "2026-02-17T05:53:20.116000Z",
  "file_hash": "843f618ee95a7af0009660fb4f01ff2cab33f4cccb17599b69201def7f61c20f",
  "private": false,
  "record": {
    "abstract": "The generalized Tur\\\u0027an number for $H$ of $G$, denoted by $\\ex(n,H,G)$, is the maximum number of copies of $H$ in an $n$-vertex $G$-free graph. When $H$ is an edge, $\\ex(n,H,G)$ is the classical Tur\\\u0027an number $\\ex(n,G)$. Let $P_k$ be the path with $k$ vertices. The square of $P_k$, denoted by $P_k^2$, is obtained by joining the pairs of vertices with distance at most two in $P_k$.\n  The Tur\\\u0027an number of $P_k^2$, $\\ex(n, P_k^2)$, was determined by several researchers. When $k=3$, $P_3^2$ is the triangle and $\\ex(n, P_3^2)$ is well-known from Mantel\u0027s theorem. When $k=4$, $\\ex(n, P_4^2)$ was solved by Dirac in a more general context. When $k=5,6$, the problem was solved by Xiao, Katona, Xiao, and Zamora. For general $k \\ge 7$, the problem was solved by Yuan in a more general context.\n  Recently, Mukherjee determined the generalized Tur\\\u0027an number $\\ex(n, K_3, P_5^2)$. In this paper, we determine the exact value of $\\ex(n, K_3, P_6^2)$ and characterize all the extremal graphs for $n \\ge 11$.",
    "arxiv_id": "2601.09454",
    "authors": [
      "Yichen Wang",
      "Ervin Gy\u0151ri"
    ],
    "categories": [
      "math.CO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "title": "The maximum number of triangles in graphs without the square of a path",
    "url": "https://arxiv.org/abs/2601.09454",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "49c2d763-0664-44ad-8440-3c4c3a981272",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}