Let $k\geq 3$. Are there infinitely many $m,n$ with $m\geq n+k$ such that \[M(n,k)>M(m,k+1)?\]
Let $k\geq 3$. Are there infinitely many $m,n$ with $m\geq n+k$ such that \[M(n,k)>M(m,k+1)?\]
The answer is yes, as proved in a strong form by Cambie [Ca24].
It is easy to see that there are infinitely many solutions to $M(n,k)>M(m,k)$. If $n_k$ is the smallest $n$ with this property (for some $m$) then are there good bounds for $n_k$? Erdős writes that he could prove $n_k/k\to \infty$, but knew of no good upper bounds.
Erdős also asked the following: If $u_k$ is minimal such that $M(u_k,k)>M(u_k+1,k)$ and $t<\min(u_k,T)$ then is it true that $M(t,k)\leq M(T,k)$? Stijn Cambie and Wouter van Doorn have observed that there are many counterexamples to this with $t=u_k-1$ and $T=u_k+1$. For example, when $k=7$ we have $u_k=7$, yet $M(6,7)=M(7,7)>M(8,7)$.
See also [677].