Erdős Problem 595 (250): Is there an infinite graph G which contains no $K_4$ and is not the union of countably many triangle-free graphs?

Monotonicity: If G is a countable union of triangle-free graphs and H ≤ G (i.e., H is a subgraph of G), then H is also a countable union of triangle-free graphs.

Proof: If G = ⨆ i, G_i with each G_i triangle-free, then H = ⨆ i, H ⊓ G_i. Each H ⊓ G_i is triangle-free because it is a subgraph of G_i.