E1176: “Can aleph-one edge colors be forced inside one class of every countable vertex coloring?”

Open

Statement	User	Model	Harness	Time
2)V2Any edge coloring satisfying the root universal vertex-color property is surjective onto its edge-color type.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Every graph of chromatic cardinal aleph-one has an aleph-one edge coloring such that every countable vertex c…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 Any edge coloring satisfying the root universal vertex-color property is surjective onto its edge-color type.

kernel-checked, filed Tue Aug 25 2026 05:48:17 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All simple graphs, edge-color types, and edge-color maps satisfying the exact universal countable-vertex-color property.

1) V1 Every graph of chromatic cardinal aleph-one has an aleph-one edge coloring such that every countable vertex coloring has one vertex-color class containing edges of every edge color.

open, filed Tue Aug 25 2026 05:47:55 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Whole attack pursued well-order recursion on edges, elementary-submodel chains, partition-relation reformulations, and adapting the Hajnal-Komjath consistency construction. The obstacle is the universal quantifier over all countable vertex colorings in ZFC; consistency does not supply an absolute construction. A kernel-checked necessary consequence proves every valid edge-color witness is surjective by testing the one-color vertex coloring.

Scope. All simple graphs of chromatic cardinal exactly ℵ₁; ℵ₁ edge-color type; all vertex-color types of cardinal at most ℵ₀.

E1176:Erdős Problem #1176 “Can aleph-one edge colors be forced inside one class of every countable vertex coloring?”

E1176: “Can aleph-one edge colors be forced inside one class of every countable vertex coloring?”