Cardinal characteristics on graphs. (English) Zbl 1245.03069

Haverkamp, Nick

J. Symb. Log. 76, No. 1, 1-33 (2011).

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.)

Keywords:

cardinal characteristic; infinite path; labelled graph; bounded graph conjecture

Citations:

Zbl 0793.05121; Zbl 0793.04003

References:

[1]	DOI: 10.1002/mana.19650300106 · Zbl 0131.20904
[2]	DOI: 10.1112/jlms/49.1.16 · Zbl 0793.04003
[3]	DOI: 10.1007/978-1-4020-5764-9_7 · Zbl 1198.03058
[4]	Graph theory (2005)
[5]	Proceedings of the London Mathematical Society 73 pp 534– (1996)
[6]	DOI: 10.1007/BF02100602 · Zbl 0793.05121