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Läuchli proved in 1961 that $\mathfrak{n}$ is finite if and only if $2^{2^{\mathfrak{n}}}＜2^{2^{\mathfrak{n}}}\cdot2$. As a consequence, $\mathfrak{n}$ is finite if and only if $2^{2^{2^{\mathfrak{n}}}}＜(2^{2^{2^{\mathfrak{n}}}})^2$. It is still open (asked by Läuchli) whether $\mathfrak{n}$ is finite if and only if $2^{2^{\mathfrak{n}}}＜(2^{2^{\mathfrak{n}}})^2$.