{
  "id": "1612.08124",
  "version": "v1",
  "published": "2016-12-24T01:50:25.000Z",
  "updated": "2016-12-24T01:50:25.000Z",
  "title": "Resilience of ranks of higher inclusion matrices",
  "authors": [
    "Rafael Plaza",
    "Qing Xiang"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $n \\geq r \\geq s \\geq 0$ be integers and $\\mathcal{F}$ a family of $r$-subsets of $[n]$. Let $W_{r,s}^{\\mathcal{F}}$ be the higher inclusion matrix of the subsets in ${\\mathcal F}$ vs. the $s$-subsets of $[n]$. When $\\mathcal{F}$ consists of all $r$-subsets of $[n]$, we shall simply write $W_{r,s}$ in place of $W_{r,s}^{\\mathcal{F}}$. In this paper we prove that the rank of the higher inclusion matrix $W_{r,s}$ over an arbitrary field $K$ is resilient. That is, if the size of $\\mathcal{F}$ is \"close\" to ${n \\choose r}$ then $\\mbox{rank}_{K}(W_{r,s}^{\\mathcal{F}}) = \\mbox{rank}_{K}(W_{r,s})$, where $K$ is an arbitrary field. Furthermore, we prove that the rank (over a field $K$) of the higher inclusion matrix of $r$-subspaces vs. $s$-subspaces of an $n$-dimensional vector space over $\\mathbb{F}_q$ is also resilient if ${\\rm char}(K)$ is coprime to $q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-24T01:50:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "higher inclusion matrix",
      "arbitrary field",
      "resilience",
      "dimensional vector space",
      "simply write"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}