Let $\kappa \leq \lambda$ cardinals with $\lambda$ infinite, and $[\lambda]^\kappa=\{Y\subseteq\lambda : ot(Y,\in)=\kappa\}$. I want to show that $[\lambda]^\kappa \asymp\ ^\kappa\lambda$.

I've aldredy proved that $[\lambda]^\kappa \asymp \{f \in \ ^\kappa\lambda: f\ \text{is increasing}\}$, and so easily $[\lambda]^\kappa\preceq \ ^\kappa\lambda$.

For the other injective function I don't have any ideas: tryng to associate the function to its range dosen't work, because I find only subset with cardinality $\kappa$ but not with order type $\kappa$.

Do you have some suggestions?


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