Cross-intersecting families of vectors

János Pach, Gábor Tardos

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

Given a sequence of positive integers p = (p1,…, pn), let Sp denote the family of all sequences of positive integers x = (x1,…, xn) such that xi ≤ pi for all i. Two families of sequences (or vectors), A, B ⊆ Sp, are said to be r-cross-intersecting if no matter how we select x ∈ A and y ∈ B, there are at least r distinct indices i such that xi = yi. We determine the maximum value of |A| · |B| over all pairs of r-cross-intersecting families and characterize the extremal pairs for r ≥ 1, provided that min pi > r + 1. The case min pi ≤ r + 1 is quite different. For this case, we have a conjecture, which we can verify under additional assumptions. Our results generalize and strengthen several previous results by Berge, Frankl, Füredi, Livingston, Moon, and Tokushige, and answers a question of Zhang. The special case r = 1 has also been settled recently by Borg.

Original languageEnglish
Title of host publicationDiscrete and Computational Geometry and Graphs - 16th Japanese Conference, JCDCGG 2013, Revised Selected Papers
EditorsHiro Ito, Toshinori Sakai, Jin Akiyama
PublisherSpringer Verlag
Pages122-137
Number of pages16
ISBN (Electronic)9783319132860
DOIs
Publication statusPublished - Jan 1 2014

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume8845
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Pach, J., & Tardos, G. (2014). Cross-intersecting families of vectors. In H. Ito, T. Sakai, & J. Akiyama (Eds.), Discrete and Computational Geometry and Graphs - 16th Japanese Conference, JCDCGG 2013, Revised Selected Papers (pp. 122-137). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8845). Springer Verlag. https://doi.org/10.1007/978-3-319-13287-711