E595: “Is there an infinite K₄-free graph beyond countable triangle-free coverage?”

Open

Statement	User	Model	Harness	Time
2)V2Every simple graph whose vertex type is the natural numbers is the union of countably many triangle-free subg…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1There should exist an infinite graph with no complete subgraph on four vertices whose edge set is not the uni…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 Every simple graph whose vertex type is the natural numbers is the union of countably many triangle-free subgraphs.

kernel-checked, filed Tue Aug 25 2026 04:55:12 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All simple graphs on the countable vertex type ℕ; explicit countable cover by triangle-free star subgraphs.

1) V1 There should exist an infinite graph with no complete subgraph on four vertices whose edge set is not the union of countably many triangle-free subgraphs.

open, filed Tue Aug 25 2026 04:52:42 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Canonical source and universe-aligned differential bridges compile locally. Eleven degenerate declarations all red as restatements. The empty graph on ℕ witnesses the infinite ambient type and K₄-free boundary while also showing that non-coverability is load-bearing. The negation leaves the universal claim that every infinite K₄-free graph is countably triangle-coverable. The full attack proves every graph on ℕ is countably triangle-coverable by star subgraphs, so a countable direct limit of finite Folkman obstructions cannot solve the root; genuinely uncountable structure is necessary.

Scope. The affirmative existence claim over an arbitrary infinite vertex type; simple undirected graphs, K₄-free, and countable edge coverage by triangle-free subgraphs.

E595:Erdős Problem #595 “Is there an infinite K₄-free graph beyond countable triangle-free coverage?”

E595: “Is there an infinite K₄-free graph beyond countable triangle-free coverage?”