Cross-intersecting families of vectors

János Pach, G. Tardos

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Given a sequence of positive integers p = (pi,…,pn), let Spdenote the family of all sequences of positive integers x = (xi,…,xn) such that xi <pi for all i. Two families of sequences (or vectors), A, B C 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 minpi > r + 1. The case minpi <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, Fu¨redi, 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
PublisherSpringer Verlag
Pages122-137
Number of pages16
Volume8845
ISBN (Print)9783319132860
DOIs
Publication statusPublished - 2014
Event16th Japan Conference on Discrete and Computational Geometry and Graphs, JCDCGG 2013 - Tokyo, Japan
Duration: Sep 17 2013Sep 19 2013

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume8845
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other16th Japan Conference on Discrete and Computational Geometry and Graphs, JCDCGG 2013
CountryJapan
CityTokyo
Period9/17/139/19/13

Fingerprint

Intersecting Family
Moon
Integer
Pi
Verify
Generalise
Family

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

Cite this

Pach, J., & Tardos, G. (2014). Cross-intersecting families of vectors. In Discrete and Computational Geometry and Graphs - 16th Japanese Conference, JCDCGG 2013, Revised Selected Papers (Vol. 8845, 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/10.1007/978-3-319-13287-7.11

Cross-intersecting families of vectors. / Pach, János; Tardos, G.

Discrete and Computational Geometry and Graphs - 16th Japanese Conference, JCDCGG 2013, Revised Selected Papers. Vol. 8845 Springer Verlag, 2014. p. 122-137 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8845).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Pach, J & Tardos, G 2014, Cross-intersecting families of vectors. in Discrete and Computational Geometry and Graphs - 16th Japanese Conference, JCDCGG 2013, Revised Selected Papers. vol. 8845, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8845, Springer Verlag, pp. 122-137, 16th Japan Conference on Discrete and Computational Geometry and Graphs, JCDCGG 2013, Tokyo, Japan, 9/17/13. https://doi.org/10.1007/978-3-319-13287-7.11
Pach J, Tardos G. Cross-intersecting families of vectors. In Discrete and Computational Geometry and Graphs - 16th Japanese Conference, JCDCGG 2013, Revised Selected Papers. Vol. 8845. Springer Verlag. 2014. p. 122-137. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-13287-7.11
Pach, János ; Tardos, G. / Cross-intersecting families of vectors. Discrete and Computational Geometry and Graphs - 16th Japanese Conference, JCDCGG 2013, Revised Selected Papers. Vol. 8845 Springer Verlag, 2014. pp. 122-137 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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