On rainbow arithmetic progressions
type: Item
instance of: scholarly article
title: On rainbow arithmetic progressions
zbMATH Open document ID: 1060.11005
author: Maria Axenovich
author: Dmitry G. Fon-Der-Flaass
published in: The Electronic Journal of Combinatorics
publication date: 2004-02-05T00:00:00Z
full work available at URL: https://eudml.org/doc/124432
full work available at URL: v11i1r1.html
review text: The authors study a variant of the celebrated theorem of van der Waerden on arithmetic progressions: coloring the set \(\{1,2,\dots , n\}\) by \(r\) colors, can we find an arithmetic progression with length \(k\) so that all its elements are colored in distinct colors? In the main part of the paper the authors investigate the case \(k=r=3\). They prove a conjecture of \textit{V. Jungić} [Exp. Math. 9, No. 3, 467--471 (2000; Zbl 0965.05093)], showing if each color class has cardinality at least \({n+4\over 6}\) then there exists a three-term arithmetic progression with distinct colors. Furthermore for \(k\geq 5\) they construct a coloring of \(\{1,2,\dots , n\}\), \(n=k(2m)\), for which each color class is \(n/k\) and there is no \(k\)-term arithmetic progression with distinct colors. In the case \(k=4\) it is constructed a coloring of \(\{1,2,\dots , n\}\), \(n=10m+1\), with the smallest color class of size \((n-1)/5\) and there is no 4-term arithmetic progression with distinct colors.
reviewed by: Norbert Hegyvári
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 05D10
zbMATH DE Number: 2038347
zbMATH Keywords: coloring of integers
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2038347