A Multidimensional Generalization of the Erdos-Szekeres Lemma on Monotone Subsequences

Tibor Szabó, Gábor Tardos

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We consider an extension of the Monotone Subsequence lemma of Erdos and Szekeres in higher dimensions. Let v1,...,vn ∈ ℝd be a sequence of real vectors. For a subset I ⊆ [n] and vector c ∈ {0, 1}d we say that I is c-free if there are no i < j ∈ I, such that, for every k = 1,...,d, vik < vjk if and only if ck = 0. We construct sequences of vectors with the property that the largest c-free subset is small for every choice of c. In particular, for d = 2 the largest c-free subset is O(n5/8) for all the four possible c. The smallest possible value remains far from being determined. We also consider and resolve a simpler variant of the problem.

Original languageEnglish
Pages (from-to)557-565
Number of pages9
JournalCombinatorics Probability and Computing
Volume10
Issue number6
DOIs
Publication statusPublished - 2001

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistics and Probability
  • Computational Theory and Mathematics
  • Applied Mathematics

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