An uncountably chromatic triple system
type: Item
instance of: scholarly article
title: An uncountably chromatic triple system
zbMATH Open document ID: 1199.03030
published in: Acta Mathematica Hungarica
publication date: 2009-12-29T00:00:00Z
review text: A triple system is a pair \((V,E)\) with \(E \subseteq [V]^3\). Its chromatic number \(\text{Chr}(V,E)\) is the least cardinal \(\mu\) such that there exists a function \(f : V \rightarrow \mu\) such that for each \(\{ x,y,z \} \in E\) we have \(| \{ f(x), f(y), f(z) \}| \geq 2\). For \(3 \leq n < \omega\), \(\mathcal{C}_n\) is the triple system \((V,E)\) with \(V = \{ x_i, y_i : i<n \}\) and \(E = \{\{x_i,y_i,x_{i+1} \} : i<n \}\) (with \(x_n = x_0\)), and \(\mathcal{T}_0\) is the triple system with vertex set \(\{ x,y,z,t\}\) and triples \(\{ x,y,z\}\), \(\{x,y,t\}\). In an earlier paper, \textit{A. Hajnal} and \textit{P. Komjáth} [Acta Math. Hung. 119, No. 1--2, 1--13 (2008; Zbl 1199.03029)] showed: If an uncountably chromatic triple system omits \(\mathcal{T}_0\), then it contains all odd circuits of length \(\geq 7\). Here the author shows that it is consistent that GCH holds and there is a triple system \(\mathcal {H}\) with \(| \mathcal {H}| = \aleph_2\), \(\text{Chr}(\mathcal {H}) = \aleph_1\) and \(\mathcal {H}\) does not contain \(\mathcal {T}_0\), \(\mathcal {C}_3\) or \(\mathcal {C}_5\).
reviewed by: Martin Weese
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 05B07
Mathematics Subject Classification ID: 05C15
zbMATH DE Number: 5651918
zbMATH Keywords: infinite graphs
zbMATH Keywords: forcing
zbMATH Keywords: chromatic number
author: Péter Komjáth
describes a project that uses: SCHOL
MaRDI profile type: Publication
full work available at URL: s10474-008-7179-6
OpenAlex ID: W2088885428
cites work: Graph Theory and Probability
cites work: Q4087192
cites work: On chromatic number of graphs and set-systems
cites work: A partition calculus in set theory
cites work: Partition Relations Connected with the Chromatic Number of Graphs
cites work: Obligatory subsystems of triple systems
cites work: Some remarks on obligatory subsystems of uncountably chromatic triple systems
DOI: 10.1007/S10474-008-7179-6
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description: scientific article; zbMATH DE number 5651918