On the number of monotone sequences

Wojciech Samotij; Benny Sudakov

doi:10.1016/j.jctb.2015.05.008

On the number of monotone sequences

Wojciech Samotij, Benny Sudakov

School of Mathematical Sciences

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

One of the most classical results in Ramsey theory is the theorem of Erdos and Szekeres from 1935, which says that every sequence of more than k² numbers contains a monotone subsequence of length k+1. We address the following natural question motivated by this result: Given integers k and n with n≥k²+1, how many monotone subsequences of length k+1 must every sequence of n numbers contain? We answer this question precisely for all sufficiently large k and n≤k²+ck^3/2/log k, where c is some absolute positive constant.

Original language	English
Pages (from-to)	132-163
Number of pages	32
Journal	Journal of Combinatorial Theory. Series B
Volume	115
DOIs	https://doi.org/10.1016/j.jctb.2015.05.008
State	Published - 1 Nov 2015

Keywords

Erdos-Rademacher
Erdos-Szekeres
Monotone subsequences
Supersaturation

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1016/j.jctb.2015.05.008

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title = "On the number of monotone sequences",

abstract = "One of the most classical results in Ramsey theory is the theorem of Erdos and Szekeres from 1935, which says that every sequence of more than k2 numbers contains a monotone subsequence of length k+1. We address the following natural question motivated by this result: Given integers k and n with n≥k2+1, how many monotone subsequences of length k+1 must every sequence of n numbers contain? We answer this question precisely for all sufficiently large k and n≤k2+ck3/2/log k, where c is some absolute positive constant.",

keywords = "Erdos-Rademacher, Erdos-Szekeres, Monotone subsequences, Supersaturation",

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AU - Samotij, Wojciech

AU - Sudakov, Benny

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N2 - One of the most classical results in Ramsey theory is the theorem of Erdos and Szekeres from 1935, which says that every sequence of more than k2 numbers contains a monotone subsequence of length k+1. We address the following natural question motivated by this result: Given integers k and n with n≥k2+1, how many monotone subsequences of length k+1 must every sequence of n numbers contain? We answer this question precisely for all sufficiently large k and n≤k2+ck3/2/log k, where c is some absolute positive constant.

AB - One of the most classical results in Ramsey theory is the theorem of Erdos and Szekeres from 1935, which says that every sequence of more than k2 numbers contains a monotone subsequence of length k+1. We address the following natural question motivated by this result: Given integers k and n with n≥k2+1, how many monotone subsequences of length k+1 must every sequence of n numbers contain? We answer this question precisely for all sufficiently large k and n≤k2+ck3/2/log k, where c is some absolute positive constant.

KW - Erdos-Rademacher

KW - Erdos-Szekeres

KW - Monotone subsequences

KW - Supersaturation

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DO - 10.1016/j.jctb.2015.05.008

M3 - مقالة

SN - 0095-8956

VL - 115

SP - 132

EP - 163

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

ER -

On the number of monotone sequences

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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