Hajnal, András; Komjáth, P.; Soukup, L.; Szalkai, I. 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 Cited in 6 Documents MSC: 05C15 Coloring of graphs and hypergraphs 03E05 Other combinatorial set theory 05C40 Connectivity Keywords:connectivity; complete graph; coloring; decomposition Citations:Zbl 0673.00009 PDFBibTeX XML