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Decompositions of edge colored infinite complete graphs. (English) Zbl 0697.05028

Combinatorics, Proc. 7th Hung. Colloq., Eger/Hung. 1987, Colloq. Math. Soc. János Bolyai 52, 277-280 (1988).
[For the entire collection see Zbl 0673.00009.]
Let K(\(\kappa)\) denote the complete graph on \(\kappa\). A \(\mu\)-coloring of a graph \(G=(V,E)\) is a function f: \(E\to \mu\) and a \(\tau\)- decomposition for a \(\mu\)-coloring is a partition p: \(V\to \tau\) such that for every \(\tau '<\tau\) there is a color \(i<\mu\) with the graph \(G'=(p^{-1}(\tau '),f^{-1}(i))\) connected. In the paper under review it is shown that every \(\mu\)-coloring of K(\(\kappa)\) admits a \(\mu\)- decomposition if \(\mu\) is finite. If \(\mu\) is infinite, a counterexample for this property exists if and only if \(cf(\kappa)=\mu^+\) (with GCH assumed).
Reviewer: K.Engel

MSC:

05C15 Coloring of graphs and hypergraphs
03E05 Other combinatorial set theory
05C40 Connectivity

Citations:

Zbl 0673.00009