All Random Solved Random Open
For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $\phi(m+1),\ldots,\phi(m+k)$ with $m+k\leq n$. Is it true that \[F(n)=(c+o(1))\log\log\log n\] for some constant $c$? Is the first pattern which fails to appear always \[\phi(m+1)>\phi(m+2)>\cdots \phi(m+k)?\] Is it true that 'natural' ordering which mimics what happens to $\phi(1),\ldots,\phi(k)$ is the most likely to appear?
Erdős [Er36b] proved that \[F(n)\asymp \log\log\log n,\] and similarly if we replace $\phi$ with $\sigma$ or $\tau$ or $\nu$ or any 'decent' additive or multiplicative function.