Problem #293: Let $k\geq 1$ and let $v(k)$ be the minimal integer which...

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Progress

Results of Bleicher and Erdős [BlEr75] imply $v(k) \gg k!$. It may be that $v(k)$ grows doubly exponentially in $\sqrt{k}$ or even $k$.

An elementary inductive argument shows that $n_k\leq ku_k$ where $u_1=1$ and $u_{i+1}=u_i(u_i+1)$, and hence\[v(k) \leq kc_0^{2^k},\]where\[c_0=\lim_n u_n^{1/2^n}=1.26408\cdots\]is the 'Vardi constant' (small improvements on this are possible as in [148]).

van Doorn and Tang [vDTa25b] have proved that\[v(k)\geq e^{ck^2}\]for some constant $c>0$, and noted a close connection to [304]. In particular, if $N(b)\ll \log\log b$ as in [304] then it is likely the methods of [vDTa25b] prove $v(k) \geq e^{e^{ck}}$ for some $c>0$.