Infinite \(K_{4}\)-free graphs are the union of triangle-free countable graphs

A problem on the decomposition of graphs ($250, proposed by Erdös and Hajnal [1]):
Is there an infinite graph \( G\) which contains no \( K_4\) and which is not the union of \( \aleph_0\) graphs which are triangle-free?

Shelah [2] proved that the existence of such a graph is consistent but it is not known if this is provable in ZFC (also see 1).


Bibliography
1 P. Erdös and A. Hajnal, On decomposition of graphs, Acta Math. Acad. Sci. Hungar. 18 (1967), 359-377.
2 S. Shelah, Consistency of positive partition theorems for graphs and models, in Set Theory and Applications, Springer Lecture Notes 1401, (Toronto, ON, 1987), 167-193.