A problem of Erd\H{o}s and Hajnal. Folkman \cite{Fo70} and Ne\v{s}et\v{r}il and R\"{o}dl \cite{NeRo75} have proved that for every $n\geq 1$ there is a graph $G$ which contains no $K_4$ and is not the union of $n$ triangle-free graphs.

See also [582] and [596].


References


[Fo70] Folkman, Jon, Graphs with monochromatic complete subgraphs in every edge
coloring. SIAM J. Appl. Math. (1970), 19-24.

[NeRo75] Ne\u set\u ril, Jaroslav and R\"odl, Vojt\v ech, Type theory of partition properties of graphs. (1975), 405-412.