**Problem**Does there exist a graph with no subgraph isomorphic to which cannot be expressed as a union of triangle free graphs?

Shelah [S] has proved that the existence of such a graph is consistent with ZFC.

## Bibliography

*[EH] P. Erdos and A. Hajnal, On decomposition of graphs, Acta Math. Acad. Sci. Hungar. 18 (1967), 359–377.

[S] 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.

* indicates original appearance(s) of problem.