Authors: | Todorčević, Stevo Tyros, Konstantinos |

Title: | Subsets of products of finite sets of positive upper density |

Journal: | Journal of Combinatorial Theory. Series A |

Volume: | 120 |

Issue: | 1 |

First page: | 183 |

Last page: | 193 |

Issue Date: | 1-Jan-2013 |

Rank: | M21 |

ISSN: | 0097-3165 |

DOI: | 10.1016/j.jcta.2012.07.010 |

Abstract: | In this note we prove that for every sequence (mq)q of positive integers and for every real 0<δ≤1 there is a sequence (nq)q of positive integers such that for every sequence (Hq)q of finite sets such that |H q|=n q for every q∈N and for every Dkq=0k-1Hq with the property thatlimsupk|D∩q=0k-1Hq||q=0k-1Hq|δ there is a sequence (Jq)q, where J qH q and |J q|=m q for all q, such that q=0k-1JqD 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 (nq)q in terms of the sequence of (mq)q. |

Keywords: | Density | Finite sets | Ramsey theory |

Publisher: | Elsevier |

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