Are there infinite cardinals $\alpha < \beta$ and a function $f:\beta\to\alpha$ with the following property?
For any $S\subseteq \beta$ with $|S| =\alpha$ the restriction $f|_S:S\to \alpha$ is injective.
Are there infinite cardinals $\alpha < \beta$ and a function $f:\beta\to\alpha$ with the following property?
For any $S\subseteq \beta$ with $|S| =\alpha$ the restriction $f|_S:S\to \alpha$ is injective.
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This is impossible. Since $f$ cannot be injective, as $\alpha<\beta$, there will be two points with the same $f$ value, $f(x)=f(y)$, and if we place those two points $x$, $y$ into an $S$ of size $\alpha$, we cannot have $f\upharpoonright S$ injective.