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Found problems: 1

2000 Miklós Schweitzer, 1
Tags: function , ordinals
Prove that there exists a function $f\colon [\omega_1]^2 \rightarrow \omega _1$ such that (i) $f(\alpha, \beta)< \mathrm{min}(\alpha, \beta)$ whenever $\mathrm{min}(\alpha,\beta)>0$; and (ii) if $\alpha_0<\alpha_1<\ldots<\alpha_i<\ldots<\omega_1$ then $\sup\left\{ a_i \colon i<\omega \right\} =\sup \left\{ f(\alpha_i, \alpha_j)\colon i,j<\omega\right\}$.