On infinite antichains of matroids
type: Item
instance of: scholarly article
title: On infinite antichains of matroids
zbMATH Open document ID: 0820.05013
author: Guoli Ding
author: Bogdan Oporowski
author: James Oxley
published in: Journal of Combinatorial Theory. Series B
publication date: 1995-07-02T00:00:00Z
review text: Robertson and Seymour have shown that there is no infinite set of graphs in which no member is a minor of another. By contrast, it is well known that the class of all matroids does contains such infinite antichains. However, for many classes of matroids, even the class of binary matroids, it is not known whether or not the class contains an infinite antichain. In this paper, we examine a class of matroids of relatively simple structure: \({\mathcal M}_{a,b,c}\) consists of those matroids for which the deletion of some set of at most \(a\) elements and the contraction of some set of at most \(b\) elements results in a matroid in which every component has at most \(c\) elements. We determine precisely when \({\mathcal M}_{a,b,c}\) contains an infinite antichain. We also show that, among the matroids representable over a finite fixed field, there is no infinite antichain in a fixed \({\mathcal M}_{a,b,c}\); nor is there an infinite antichain when the circuit size is bounded.
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 767678
zbMATH Keywords: well-quasi-order
zbMATH Keywords: minor
zbMATH Keywords: matroids
zbMATH Keywords: infinite antichains
MaRDI profile type: Publication
full work available at URL: jctb.1995.1003
OpenAlex ID: W2126752065
DOI: 10.1006/JCTB.1995.1003
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description: scientific article; zbMATH DE number 767678