How to apply strong pigeonhole principle to this question?

Question

For each $r\geq 1$, let $L(r,n)$ denote the integer for which given any sequence of length $L(r,n)$, and any $r$-coloring of the elements of the sequence, there exists a monochromatic increasing sequence of length $n+1$ or a monochromatic decreasing sequence of length $n+1$, but there exists a sequence of length $L(r,n)-1$ and an $r$-coloring of that sequence which has no monochromatic increasing or decreasing sequence of length $n+1$. So, $L(1,n)=n^2+1$ for all $n$. Find $L(r,n)$ for all $r,n\geq 1$.

Answer 1

HINT: Given an $r$-coloring of $\langle a_1,\ldots,a_m\rangle$, you can consider each of the $r$ monochromatic subsequences independently. You need at least one of them to be long enough to ensure that it has a monotone subsequence of length $n+1$, and you know what $L(1,n)$ is.