Is there an infinite graph $G$ which contains no $K_{4}$ and is not the union of countably many triangle-free graphs?

Open — best to date is a honest null, not yet sealed.

graph theory · open · prize $250 · formalized (Lean) · 1 attempt

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

honest null

needs verification

attempted via frontier '?' (transfer_strength=n/a) -> no_progress

No solve/partial on this pass. Transfer into the owned frontier was 'n/a'. Do not re-attack cold; needs a new idea or richer accumulated context.

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let me rewrite your question in a standard way. Saying that

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

AMS 5 · open (literature)

theorem erdos_595 : answer(sorry) ↔
    ∃ (V : Type*) (_ : Infinite V) (G : SimpleGraph V),
      G.CliqueFree 4 ∧ ¬IsCountableUnionOfTriangleFree G

open