A note on monoton waves
type: Item
instance of: scholarly article
title: A note on monoton waves
zbMATH Open document ID: 0671.10049
DOI: 10.1016/0097-3165(89)90024-1
author: Hanno Lefmann
published in: Journal of Combinatorial Theory. Series A
publication date: 1989-01-01T00:00:00Z
review text: A sequence \(x_ 0<...<x_{k-1}\) of nonnegative integers is called (i) an ascending wave if \(x_ i-x_{i-1}\leq x_{i+1}-x_ i\) for \(0<i<k- 1\), and (ii) a descending wave if \(x_ i-x_{i-1}\geq x_{i+1}-x_ i\) for \(0<i<k-1\). For positive integers \(k,\ell \geq 2\) let \(\omega(k,\ell)\) denote the least positive integer such that for every integer \(n\geq \omega (k,\ell)\) every integral sequence \(0\leq x_ 0<...<x_{n-1}\) contains a subsequence which is either a k-term ascending wave or an \(\ell\)-term descending wave. Using the weak version of Ramsey's theorem the author proves that \[ \omega (k,\ell)=\binom{k+\ell-4}{k-2}+\ell. \]
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 05C55
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 4097505
zbMATH Keywords: discrete structures similar to arithmetic progressions
zbMATH Keywords: monotone
zbMATH Keywords: waves
zbMATH Keywords: weak Ramsey's theorem
zbMATH Keywords: ascending wave
zbMATH Keywords: descending wave
reviewed by: Štefan Porubský
MaRDI profile type: Publication
full work available at URL: 0097-3165(89)90024-1
OpenAlex ID: W2171670133
cites work: Quasi-progressions and descending waves
cites work: Q5759552
cites work: Weak versions of Ramsey's theorem
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description: scientific article; zbMATH DE number 4097505