{
  "id": "1611.06842",
  "version": "v1",
  "published": "2016-11-21T15:40:24.000Z",
  "updated": "2016-11-21T15:40:24.000Z",
  "title": "Almost tiling of the Boolean lattice with copies of a poset",
  "authors": [
    "István Tomon"
  ],
  "comment": "9 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $P$ be a partially ordered set. If the Boolean lattice $(2^{[n]},\\subset)$ can be partitioned into copies of $P$ for some positive integer $n$, then $P$ must satisfy the following two trivial conditions: (1) the size of $P$ is a power of $2$, (2) $P$ has a unique maximal and minimal element. Resolving a conjecture of Lonc, it was shown by Gruslys, Leader and Tomon that these conditions are sufficient as well. In this paper, we show that if $P$ only satisfies condition (2), we can still almost partition $2^{[n]}$ into copies of $P$. We prove that if $P$ has a unique maximal and minimal element, then there exists a constant $c=c(P)$ such that all but at most $c$ elements of $2^{[n]}$ can be covered by disjoint copies of $P$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-21T15:40:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boolean lattice",
      "minimal element",
      "unique maximal",
      "trivial conditions",
      "satisfies condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}