Symmetry and the Ramsey degree of posets
type: Item
instance of: scholarly article
title: Symmetry and the Ramsey degree of posets
zbMATH Open document ID: 0873.05076
DOI: 10.1016/S0012-365X(96)00236-1
author: Willem Louw Fouché
published in: Discrete Mathematics
publication date: 1997-10-26T00:00:00Z
review text: If \(P\) is a finite poset, let \(e(P)\) be the number of linear extensions of \(P\) and \(\Aut(P)\) the group of (order) isomorphisms of \(P\). The author defines \(t(P)= e(P)/|\Aut(P)|\) and he also shows that \(t(P)\) is an integer, the Ramsey degree of \(P\) and he observes that if \(t(P)=1\), then for each poset \(Q\) and each natural number \(r\), there is a poset \(R\) such that for every \(r\)-coloring of the set of copies of \(R\) in \(P\), there is a monochromatic copy of \(Q\) in \(R\) (per J. Nešetřil and V. Rödl). Using this interesting definition and the approach leading to the above characterization his main theorem is Theorem 1: For posets \(P\) and \(Q\) and a natural number \(r\), there exists a poset \(R\) such that for any \(r\)-coloring \(\chi\) of the copies of \(P\) in \(R\), there exists a copy \(Q'\) of \(Q\) in \(R\) such that \(\chi\) assumes at most \(t(P)\) values on the set of copies of \(Q'\) in \(P\). A copy of a poset \(A\) in a poset \(B\) is the image of \(A\) in \(B\) via an order-preserving injection.
Mathematics Subject Classification ID: 05D10
Mathematics Subject Classification ID: 06A07
zbMATH DE Number: 1018522
zbMATH Keywords: embedding
zbMATH Keywords: automorphism
zbMATH Keywords: copy
zbMATH Keywords: coloring
zbMATH Keywords: poset
zbMATH Keywords: linear extensions
zbMATH Keywords: Ramsey degree
reviewed by: J. Neggers
MaRDI profile type: Publication
cites work: Ramsey's Theorem for n-Parameter Sets
cites work: Q3997075
cites work: Partitions of finite relational and set systems
cites work: Combinatorial partitions of finite posets and lattices - Ramsey lattices
cites work: Q3680868
full work available at URL: s0012-365x(96)00236-1
OpenAlex ID: W2092537246
description: scientific article; zbMATH DE number 1018522