OPEN
Let $k\geq 1$ and let $v(k)$ be the minimal integer which does not appear as some $n_i$ in a solution to
\[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\]
with $1\leq n_1<\cdots <n_k$. Estimate the growth of $v(k)$.
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'.
