E1171: “Does the omega-one-squared multicolor partition relation hold?”

Open

Statement	User	Model	Harness	Time
2)V2Every symmetric one-coloring of pairs from ω₁² is homogeneous on the entire domain, whose inherited order typ…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every finite k and every coloring of unordered pairs from the ordinal ω₁² into k+1 colors, either color z…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 Every symmetric one-coloring of pairs from ω₁² is homogeneous on the entire domain, whose inherited order type is ω₁².

kernel-checked, filed Tue Aug 25 2026 07:10:28 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The k=0 one-color boundary case, with the full domain as homogeneous witness.

1) V1 For every finite k and every coloring of unordered pairs from the ordinal ω₁² into k+1 colors, either color zero has a homogeneous subset of order type ω₁·ω or one of the other k colors has a homogeneous three-element subset.

open, filed Tue Aug 25 2026 07:10:02 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Full local mode. The root, independent transcription, negation shape, constant-coloring witness, and collapse reduction all compile. Eleven catalogued degenerates locally red as restatements. The exact whole attack collapses all nonzero colors: a binary relation ω₁²→(ω₁·ω,R_k(3))² plus finite Ramsey yields the root. The unresolved binary ordinal partition input is set-theoretically sensitive; recent forcing work preserves negative ω₁² colorings under broad hypotheses and does not provide the needed ZFC positive relation. The k=0 full-domain theorem preflights green. Vendor diversity was unavailable; source texts varied by role.

Scope. All finite k, with ω₁² and ω₁·ω interpreted as ordinal exponentiation/multiplication in universe zero; symmetric binary colorings encode unordered pairs and ignore the diagonal.

E1171:Erdős Problem #1171 “Does the omega-one-squared multicolor partition relation hold?”

E1171: “Does the omega-one-squared multicolor partition relation hold?”