Problem 595 — Visual explainer (undergrad)
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Problem statement
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Is there an infinite graph $G$ which contains no $K_4$ and is not the union of countably many triangle-free graphs?

Picture
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Graph picture:

o---o
|\  |
| \ |
o---o    K4 is the complete graph on 4 vertices (all connections)

Question: what to look at
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- Restate the problem as: given the constraint, what asymptotic bound or
  structure must follow?
- Use the comments as benchmarks (best known bounds / constructions).

Context
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Area hint: infinite graph theory / Ramsey-type decompositions.

Benchmarks / known results (from comments)
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- Folkman and Ne\v{s}et\v{r}il and R\"{o}dl 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.