E1068: “Does every aleph-one chromatic graph contain a countable infinitely connected subgraph?”

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Statement	User	Model	Harness	Time
2)V2Every infinitely connected graph under the root definition is connected in the ordinary graph-theoretic sense.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Every graph of chromatic cardinal aleph-one contains a countable induced subgraph in which every distinct ver…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 Every infinitely connected graph under the root definition is connected in the ordinary graph-theoretic sense.

kernel-checked, filed Tue Aug 25 2026 05:31:27 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All vertex types and simple graphs; the exact pairwise internally-disjoint-path definition used by the root.

1) V1 Every graph of chromatic cardinal aleph-one contains a countable induced subgraph in which every distinct vertex pair has infinitely many pairwise internally disjoint paths.

open, filed Tue Aug 25 2026 05:31:10 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Whole attack pursued countable elementary-submodel closure, extraction of arbitrarily high finite-connectivity subgraphs, infinite Menger compactness, and transfinite coloring contradictions. The missing step is a coherent countable set supporting infinitely many internally disjoint paths for every pair; finite-k witnesses need not nest. Bowler-Pitz obstructs uncountable extraction but explicitly leaves this countable version open. A separately checked structural lemma proves the target connectivity notion implies ordinary connectedness.

Scope. All vertex types and simple graphs with chromatic cardinal exactly ℵ₁; existential countable vertex set; induced subgraph and vertex-internal path disjointness.

E1068:Erdős Problem #1068 “Does every aleph-one chromatic graph contain a countable infinitely connected subgraph?”

E1068: “Does every aleph-one chromatic graph contain a countable infinitely connected subgraph?”