Problem 595

Erdős Problem #595: declared status 'open'. Formalized: yes. Is there an infinite graph $G$ which contains no $K_4$ and is not the union of countably many triangle-free graphs? Current best: Folkman [Fo70] and Ne\v{s}et\v{r}il and R\"{o}dl [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. Prize: $250. Tags: graph theory, set theory.