On Taylor's problem (Q1912691)

Taylor (see \textit{P. Erdös}, \textit{A. Hajnal} and the second author, Topics in Topology, Colloq. Keszthely 1972, Colloquia Math. Soc. János Bolyai 8, 243-255 (1974; Zbl 0299.02083)) posed the following problem. If \(\kappa\), \(\lambda\) are uncountable cardinals and \(X\) is a \(\kappa\)-chromatic graph, is there a \(\lambda\)-chromatic graph \(Y\) such that every finite subgraph of \(Y\) appears as a subgraph of \(X\)? The present authors give some results on this conjecture of Taylor, in the case where \(|X|= \kappa\) and \(|Y|= \lambda\).