{
  "id": "1211.3948",
  "version": "v1",
  "published": "2012-11-16T16:38:22.000Z",
  "updated": "2012-11-16T16:38:22.000Z",
  "title": "Subsets of Products of Finite Sets of Positive Upper Density",
  "authors": [
    "Stevo Todorcevic",
    "Konstantinos Tyros"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this note we prove that for every sequence $(m_q)_{q}$ of positive integers and for every real $0<\\delta\\leqslant1$ there is a sequence $(n_q)_{q}$ of positive integers such that for every sequence $(H_q)_{q}$ of finite sets such that $|H_q|=n_q$ for every $q\\in\\mathbb{N}$ and for every $D\\subseteq \\bigcup_k\\prod_{q=0}^{k-1}H_q$ with the property that $$\\limsup_k \\frac{|D\\cap \\prod_{q=0}^{k-1} H_q|}{|\\prod_{q=0}^{k-1}H_q|}\\geqslant\\delta$$ there is a sequence $(J_q)_{q}$, where $J_q\\subseteq H_q$ and $|J_q|=m_q$ for all $q$, such that $\\prod_{q=0}^{k-1}J_q\\subseteq D$ for infinitely many $k.$ This gives us a density version of a well-known Ramsey-theoretic result. We also give some estimates on the sequence $(n_q)_{q}$ in terms of the sequence of $(m_q)_{q}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-16T16:38:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive upper density",
      "finite sets",
      "positive integers",
      "well-known ramsey-theoretic result",
      "density version"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.3948T"
    }
  }
}