OPEN - $250
Is there an infinite graph $G$ which contains no $K_4$ and is not the union of countably many triangle-free graphs?
A problem of Erdős and Hajnal. Folkman
[Fo70] and Nešetřil and Rödl
[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].