All Random Solved Random Open
Let $N\geq 1$ and let $t(N)$ be the least integer $t$ such that there is no solution to \[1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}\] with $t=n_1<\cdots <n_k\leq N$. Estimate $t(N)$.
Erdős and Graham [ErGr80] could show \[t(N)\ll\frac{N}{\log N},\] but had no idea of the true value of $t(N)$.

Solved by Liu and Sawhney [LiSa24] (up to $(\log\log N)^{O(1)}$), who proved that \[\frac{N}{(\log N)(\log\log N)^3(\log\log\log N)^{O(1)}}\ll t(N) \ll \frac{N}{\log N}.\]