$(a)$Let us consider two sequences $a =( a_1,...,a_n)$ and $b =( b_1,...,b_n)$ of distinct real numbers. Show that indices $i_1,...,i_k,1≤ i_1 < ···< i_k ≤ n$, always exist with $k = \Big\lceil n^\frac{1}{4}\Big\rceil$ such that the subsequences determined by them in both $a$ and $b$ are monotone.
$(b)$ Show that the bound for $k$ in $(a)$ cannot be improved in general.
My attempts
I think Erdos-Szekeres theorem would help, but I don't see how, since we don't know the factors of $n$. Pigeonhole principle also seems not to do.
Please help.
The factors of $n$ do not matter. Just show that there is a monotone sequence in $a_n$ of length $\lceil \sqrt n \rceil$ using the theorem. Then select that subsequence of the $b$s and do the same.