Rado numbers for \(a(x+y)bz\)
type: Item
instance of: scholarly article
title: Rado numbers for \(a(x+y)bz\)
zbMATH Open document ID: 0901.05090
author: Heiko Harborth
author: Silke Maasberg
published in: Journal of Combinatorial Theory. Series A
publication date: 1998-11-26T00:00:00Z
review text: A well-known theorem of Schur guarantees for any number of colors \(k\) the existence of a minimal integer \(S(k)\) such that any \(k\)-coloring of the numbers \(1,2,\dots, S(k)\) contains a monochromatic solution of the equation \(x+y= z\). The authors consider the more general case \(a(x+ y)= bz\). They denote by \(Ra_k(a,b)\) a generalization of the number \(S(k)\) \((Ra_k(a, a)= S(k))\). The main results of this paper are formulas describing the values of \(Ra_2(a, b)\) for \(b= 2\) and \(b= a+1\), namely: \(Ra_2(a, 2)= a(a^2+ 1)/2\) and \(Ra_2(a, a+1)= a(a+ 1)\) for \(a\geq 3\), and a computer-aided calculation of \(Ra_k(a, b)\) for some \(a,b\leq 23\), \(k= 2\) and \(a,b\leq 5\), \(k= 3\). Moreover, they consider a further generalization of the equation \(a(x+y)= bz\) to a system of \(m\) equations \(a(x_n+ x_{n+1})= bx_{n+2}\), \(n= 1,\dots,m\). Furthermore, the authors define numbers \(Ra_k(m; a,b)\) such that \(Ra_k(1; a,b)= Ra_k(a, b)\).
Mathematics Subject Classification ID: 05D10
zbMATH DE Number: 1093111
zbMATH Keywords: Rado number
zbMATH Keywords: Schur number
reviewed by: Marek Kubale
MaRDI profile type: Publication
full work available at URL: jcta.1997.2810
OpenAlex ID: W2091190125
cites work: On the degree of regularity of some equations
cites work: Q3997075
cites work: Q4883334
cites work: Q3838218
cites work: Q4304379
DOI: 10.1006/JCTA.1997.2810
description: scientific article; zbMATH DE number 1093111