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Some remarks on the simultaneous chromatic number. (English) Zbl 1048.03039

The authors present partial results, variants and consistency results concerning a yet unsolved conjecture. The conjecture is due to P. Erdős, F. Galvin and A. Hajnal [in: A. Hajnal et al. (eds.), Infinite and finite sets, Colloq. Math. Soc. János Bolyai 10, 425–513 (1975; Zbl 0324.04005)] and says that if \(X\) is a graph on the ground set \(V\) with \(\chi(X)=\aleph_1\), then \(X\) has an edge coloring \(F\) with \(\aleph_1\) colors such that if \(V\) is decomposed into \(\aleph_0\) parts then there is one part in which \(F\) takes all values.

MSC:

03E35 Consistency and independence results
05C15 Coloring of graphs and hypergraphs

Citations:

Zbl 0324.04005
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