Partition regularity of \((M,P,C)\)-systems
type: Item
instance of: scholarly article
title: Partition regularity of \((M,P,C)\)-systems
zbMATH Open document ID: 0785.05013
DOI: 10.1016/0097-3165(93)90084-L
author: Neil Hindman
author: Hanno Lefmann
published in: Journal of Combinatorial Theory. Series A
publication date: 1993-12-02T00:00:00Z
review text: A system of equations is partition regular over the set \(\mathbb{N}\) (positive integers) if, whenever \(\mathbb{N}\) is partitioned into finitely many classes, some one of those classes contains a solution to the given system. R. Rado characterized systems of homogeneous linear equations with integer coefficients which are parition regular. An alternate characterization was given in terms of \((m,p,c)\)-sets by \textit{W. Deuber} [Math. Z. 133, 109-123 (1973; Zbl 0254.05011)]. An \((M,P,C)\)-system is a choice of an \((m,p,c)\)-set for which \((m,p,c)\in \mathbb{N}^ 3\), together with all finite sums, choosing at most one from each \((m,p,c)\)-set. The authors show that \((M,P,C)\)-systems are partition regular. They also prove a multidimensional version of this, similar to a generalization of the Finite Sum Theorem proved by \textit{K. R. Millikan} [J. Comb. Theory, Ser. A 18, 276-290 (1975; Zbl 0323.05001)] and \textit{A. D. Taylor} [J. Comb. Theory, Ser. A 21, 137-146 (1976; Zbl 0341.05010)].
Mathematics Subject Classification ID: 05A18
Mathematics Subject Classification ID: 05A05
zbMATH DE Number: 446852
zbMATH Keywords: finite sum theorem
zbMATH Keywords: partition
zbMATH Keywords: homogeneous linear equations
reviewed by: Gerald L. Alexanderson
MaRDI profile type: Publication
cites work: Partitionen und lineare Gleichungssysteme
cites work: Partitions and sums of \((m,p,c)\)-sets
cites work: Q5589084
cites work: Q3997075
cites work: Q3205333
cites work: Ramsey's theorem with sums or unions
cites work: Studien zur Kombinatorik
cites work: A canonical partition relation for finite subsets of \(\omega\)
description: scientific article; zbMATH DE number 446852