{
  "id": "1910.03740",
  "version": "v1",
  "published": "2019-10-09T01:35:06.000Z",
  "updated": "2019-10-09T01:35:06.000Z",
  "title": "The Resolution of Keller's Conjecture",
  "authors": [
    "Joshua Brakensiek",
    "Marijn Heule",
    "John Mackey"
  ],
  "comment": "18 pages, 4 figures, 3 tables",
  "categories": [
    "math.CO",
    "cs.DM",
    "cs.LO",
    "math.MG"
  ],
  "abstract": "We consider two graphs, $G_{7,3}$ and $G_{7,4}$, related to Keller's conjecture in dimension 7. We show, with computer assistance, that every maximal clique in either graph contains a facesharing pair of vertices. Doing so shows that every unit cube tiling of $\\mathbb{R}^7$ contains a facesharing pair of cubes. Since there is a faceshare-free unit cube tiling of $\\mathbb{R}^8$, this completely resolves Keller's conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-09T01:35:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "resolution",
      "resolves kellers conjecture",
      "facesharing pair",
      "maximal clique",
      "graph contains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}