{
  "id": "2109.09925",
  "version": "v1",
  "published": "2021-09-21T02:57:32.000Z",
  "updated": "2021-09-21T02:57:32.000Z",
  "title": "Towards supersaturation for oddtown and eventown",
  "authors": [
    "Jason O'Neill"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a collection $\\mathcal{A}$ of subsets of an $n$ element set, let $\\text{op}(\\mathcal{A})$ denote the number of distinct pairs $A,B \\in \\mathcal{A}$ for which $|A \\cap B|$ is odd. Using linear algebra arguments, we prove that for any collection $\\mathcal{A}$ of $2^{\\lfloor n/2 \\rfloor}+1$ even-sized subsets of an $n$ element set that $\\text{op}(\\mathcal{A}) \\geq 2^{\\lfloor n/2 \\rfloor-1}$. We also prove that for any collection $\\mathcal{A}$ of $n+1$ odd-sized subsets of an $n$ element set that $\\text{op}(\\mathcal{A}) \\geq 3$, and that both of these results are best possible. We also consider the problem for larger collections of odd-sized and even-sized sets respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-21T02:57:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "element set",
      "supersaturation",
      "linear algebra arguments",
      "distinct pairs",
      "larger collections"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}