Some remarks on the simultaneous chromatic number (Q1878592)

The authors present partial results, variants and consistency results concerning a yet unsolved conjecture. The conjecture is due to \textit{P. Erdős, F. Galvin} and \textit{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.