FALSE: every list of $mn$ pairwise distinct reals has a strictly increasing sublist of length $m+1$ or a strictly decreasing sublist of length $n+1$

FALSE. The statement

for all $m, n \in \mathbb{N}$, every pairwise distinct finite list of reals of length $mn$ has a strictly increasing sublist of length $m+1$ or a strictly decreasing sublist of length $n+1$ (A finite list of reals, and its strictly increasing and strictly decreasing sublists).

This is Every list of $mn+1$ pairwise distinct reals has a strictly increasing sublist of length $m+1$ or a strictly decreasing sublist of length $n+1$ with the length lowered from $mn+1$ to $mn$. The true theorem is sharp, so lowering the length by one destroys it, and it does so at every pair $(m,n)$ rather than at some exceptional pair.

• The false statement is not weaker in one place and true in another. The witness of For all $m$ and $n$ there is a list of $mn$ pairwise distinct reals with no strictly increasing sublist of length $m+1$ and no strictly decreasing sublist of length $n+1$ exists for every pair of naturals, so there is no range of $m$ and $n$ in which the lowered bound holds.

• The degenerate pairs fail too, and for a different reason. At $m = 0$ or $n = 0$ the length $mn$ is $0$, the list is empty, and it has no sublist of any positive length; both alternatives ask for a sublist of length at least $1$. At those pairs the claim fails without any construction being needed.