Erdős Problem 415

### What is the conjecture

https://www.erdosproblems.com/415

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?

Status: open

### Choose either option
- [ ] I plan on working on this conjecture
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else