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Miklós Schweitzer 2000, Problem 1

Source: Miklós Schweitzer 2000

July 30, 2016
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Problem Statement

Prove that there exists a function f ⁣:[ω1]2→ω1f\colon [\omega_1]^2 \rightarrow \omega _1 such that (i) f(α,β)<min(α,β)f(\alpha, \beta)< \mathrm{min}(\alpha, \beta) whenever min(α,β)>0\mathrm{min}(\alpha,\beta)>0; and (ii) if α0<α1<…<αi<…<ω1\alpha_0<\alpha_1<\ldots<\alpha_i<\ldots<\omega_1 then sup⁡{ai ⁣:i<ω}=sup⁡{f(αi,αj) ⁣:i,j<ω}\sup\left\{ a_i \colon i<\omega \right\} =\sup \left\{ f(\alpha_i, \alpha_j)\colon i,j<\omega\right\}.
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