### Abstract

We consider an extension of the Monotone Subsequence lemma of Erdos and Szekeres in higher dimensions. Let v^{1},...,v^{n} ∈ ℝ^{d} 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, v^{i}_{k} < v^{j}_{k} if and only if c^{→}_{k} = 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(n^{5/8}) 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 language | English |
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Pages (from-to) | 557-565 |

Number of pages | 9 |

Journal | Combinatorics Probability and Computing |

Volume | 10 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2001 |

### ASJC Scopus subject areas

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