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Cardinal characteristics on graphs. (English) Zbl 1245.03069
Summary: A cardinal characteristic can often be described as the smallest size of a family of sequences which has a given property. Instead of this traditional concern for a smallest realization of the given property, a basically new approach, taken in [R. Diestel and I. Leader, Invent. Math. 108, No. 1, 131–162 (1992; Zbl 0793.05121); R. Diestel, S. Shelah and J. Steprāns, J. Lond. Math. Soc., II. Ser. 49, No. 1, 16–24 (1994; Zbl 0793.04003)], asks for a realization whose members are sequences of labels that correspond to 1-way infinite paths in a labelled graph. We study this approach as such, establishing tools that are applicable to all these cardinal characteristics. As an application, we demonstrate the power of the tools developed by presenting a short proof of the bounded graph conjecture [Diestel and Leader, loc. cit.].
MSC:
03E17 Cardinal characteristics of the continuum
05C38 Paths and cycles
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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