E70: “Does the continuum satisfy every countable ordinal versus finite triple partition relation?”

Open

Statement	User	Model	Harness	Time
15)V2Any counterexample to the red ω·2 versus blue-four conclusion is hereditarily mixed: every uncountable vertex…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
14)V2Every finite anchor partition of an uncountable set has an uncountable single-signature child.	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
13)V2If no finite anchor canonizes a triple coloring, then every finite anchor admits two coordinatewise signature…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
12)V2If colors of triples are determined by the finite signatures recording each vertex’s colors against pairs fro…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
11)V2Every triple colouring of the continuum determined by finitely many vertex types has either a red subset of o…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
10)V2There is a coherent pivotCover colouring on the lexicographic two-block carrier Bool × ℕ of order type ω·2 wh…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
9)V2A three-vertex graph is a complete-bipartite cut exactly when it avoids the odd link triangles with one or th…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
8)V2Any symmetric triple colouring of the continuum whose pivot links are complete bipartite graphs—even with an…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
7)V2If every pivot link is the same complete-bipartite cut and the links come from one symmetric triple colouring…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
6)V2There is an explicit ω·2-ordered family of complete-bipartite red link graphs that hits every canonical two-b…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
5)V2Every pivotCover colouring either already has a red ω·2 order-copy or every pivot-link red graph meets every…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
4)V2The first open relation (𝔠).ord → (ω·2,4)₂³ is equivalent to proving a red ω·2 order-copy for every symmetri…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
3)V2For every countable ordinal β, the continuum satisfies (𝔠).ord → (β,3)₂³.	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
2)V2For every countable ordinal β, the continuum satisfies (𝔠).ord → (β,2)₂³.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every countable ordinal β and every finite n at least 2, every two-colouring of triples from the continuu…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

15) V2 Any counterexample to the red ω·2 versus blue-four conclusion is hereditarily mixed: every uncountable vertex subset contains distinct triples of both colors.

kernel-checked, filed Tue Aug 25 2026 06:21:34 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All triple predicates on the continuum, conditional on absence of both a red ordered ω·2 copy and a blue four-set.

14) V2 Every finite anchor partition of an uncountable set has an uncountable single-signature child.

kernel-checked, filed Tue Aug 25 2026 06:18:19 GMT+0000 (Coordinated Universal Time) by @woshuajolk

If every uncountable subfiber is mixed, that one child already contains a split witness, so no simultaneous large siblings or continuum-cofinality assumption is needed at a finite stage.

Scope. All ternary predicates on arbitrary types, finite anchors, and uncountable hereditarily mixed vertex sets.

13) V2 If no finite anchor canonizes a triple coloring, then every finite anchor admits two coordinatewise signature-indistinguishable distinct triples with opposite colors.

kernel-checked, filed Tue Aug 25 2026 05:59:42 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The statement also isolates continuumSplitAt, the strictly stronger large-fiber condition needed for a König/Cantor limit argument.

Scope. The exact one-step contrapositive splitting theorem for finite-anchor canonization, plus an explicit formulation of the missing continuum-sized-fiber strengthening.

12) V2 If colors of triples are determined by the finite signatures recording each vertex’s colors against pairs from one finite anchor set, then the coloring factors through finitely many vertex types.

kernel-checked, filed Tue Aug 25 2026 05:46:27 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Thus the precisely stated anchorSplitPrinciple reduces every symmetric pivotCover coloring to the solved finite-type case or the Ramsey conclusion.

Scope. A formal finite-anchor canonization criterion and exact named residual principle for the n=4, β=ω·2 continuum case.

11) V2 Every triple colouring of the continuum determined by finitely many vertex types has either a red subset of order type ω·2, with an explicit order isomorphism, or a blue four-set.

kernel-checked, filed Tue Aug 25 2026 05:39:16 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The n=4, β=ω·2 case for all continuum triple colourings factoring through a finite partition of the vertex set.

10) V2 There is a coherent pivotCover colouring on the lexicographic two-block carrier Bool × ℕ of order type ω·2 whose 2+2 obstruction type and distinguished second-block pivots are constant on every infinite thinning and compatible across all overlaps, yet every canonical two-block thinning contains a blue triple.

kernel-checked, filed Tue Aug 25 2026 05:31:30 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. An explicit polarized two-ω-block control showing that stable 2+2 witness type plus overlap compatibility alone does not force a red ω·2 copy.

9) V2 A three-vertex graph is a complete-bipartite cut exactly when it avoids the odd link triangles with one or three edges.

kernel-checked, filed Tue Aug 25 2026 05:24:03 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Under cross-pivot coherence and pivotCover, every four distinct continuum vertices exhibit such a forbidden triangle in at least one pivot link.

Scope. The exact finite forbidden-pattern reduction for symmetric pivotCover triple colourings on (𝔠).ord; it classifies three-edge cut patterns and locates an obstruction on every four-set.

8) V2 Any symmetric triple colouring of the continuum whose pivot links are complete bipartite graphs—even with an arbitrary pivot-dependent cut at every vertex—is identically blue on distinct triples.

kernel-checked, filed Tue Aug 25 2026 05:15:13 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Each pivot cut is constant off its pivot, the colouring admits a common-cut representation, and a blue four-set exists.

Scope. Symmetric triple colourings on (𝔠).ord with arbitrary Bool-valued pivot-dependent complete-bipartite link cuts.

7) V2 If every pivot link is the same complete-bipartite cut and the links come from one symmetric triple colouring, then every distinct triple is blue; in particular there is a blue four-set.

kernel-checked, filed Tue Aug 25 2026 05:03:11 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Symmetric triple colourings on (𝔠).ord whose every pivot link is defined by one common Bool-valued vertex cut.

6) V2 There is an explicit ω·2-ordered family of complete-bipartite red link graphs that hits every canonical two-block thinning but has neither a blue two-block thinning nor a red triangle; it fails exactly cross-pivot coherence.

kernel-checked, filed Tue Aug 25 2026 04:53:14 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. An explicit abstract family of link graphs on Bool × ℕ with lexicographic order type ω·2; this is a barrier to using the hitting property alone, not a triple-colouring counterexample.

5) V2 Every pivotCover colouring either already has a red ω·2 order-copy or every pivot-link red graph meets every disjoint subset of order type ω·2.

kernel-checked, filed Tue Aug 25 2026 04:44:18 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For every triple predicate on (𝔠).ord satisfying pivotCover; the conclusion is a red ω·2 order-copy or an ω·2-hitting condition for every pivot link.

4) V2 The first open relation (𝔠).ord → (ω·2,4)₂³ is equivalent to proving a red ω·2 order-copy for every symmetric colouring whose pivot link graphs have each blue triangle covered by a red triple on its non-pivot vertices.

kernel-checked, filed Tue Aug 25 2026 04:37:20 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Exactly the n = 4, β = ω·2 instance over (𝔠).ord; the reduction quantifies over every symmetric triple colouring and uses an explicit order isomorphism (ω·2).ToType ≃o s.

3) V2 For every countable ordinal β, the continuum satisfies (𝔠).ord → (β,3)₂³.

kernel-checked, filed Tue Aug 25 2026 04:05:17 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For every β : Ordinal with β.card ≤ ℵ₀, every symmetric two-colouring of distinct triples from (𝔠).ord has a red subset of order type β or a blue three-element subset.

2) V2 For every countable ordinal β, the continuum satisfies (𝔠).ord → (β,2)₂³.

kernel-checked, filed Tue Aug 25 2026 03:31:38 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For every β : Ordinal with β.card ≤ ℵ₀, every two-colouring of triples from (𝔠).ord has a red β or a blue two-element subset.

1) V1 For every countable ordinal β and every finite n at least 2, every two-colouring of triples from the continuum has either a red subset of order type β or a blue subset of size n.

open, filed Tue Aug 25 2026 03:31:16 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Formal written first and read back term by term. The continuum is its initial ordinal (𝔠).ord; β.card ≤ ℵ₀ is countability; n : ℕ with 2 ≤ n is the finite target; permutation invariance makes the predicate a colouring of unordered triples; red uses order type and blue uses cardinality. The search asymmetry is kernel-checked formalization of all quantifier and symmetry conventions, which the original notation leaves implicit.

Scope. For every β : Ordinal with β.card ≤ ℵ₀ and every n ∈ ℕ with 2 ≤ n, the partition relation (𝔠).ord → (β,n)₂³ holds.

E70:Erdős Problem #70 “Does the continuum satisfy every countable ordinal versus finite triple partition relation?”

E70: “Does the continuum satisfy every countable ordinal versus finite triple partition relation?”