The Erdős-Szekeres Theorem

Question

What is the best way to prove that the theorem is tight? In other words, how do I find a sequence of length $ n^{2} + 1 $ that does not contain any non-decreasing or non-increasing sub-sequence of length $ n + 2 $?

Answer 1

Do you mean a sequence of length $n^2 + 1$? Then you just construct it like so: $$n, n-1, n-2, \dots, 1, 2n, 2n-1, 2n-2, \dots, n+1, \dots, n^2+1, n^2, n^2-1, \dots, n^2-n$$

That is, you concatenate internally decreasing sequences of length $n$ which are externally increasing (if that makes sense). The last sequence has the extra element, and forms the largest monotonic sequence (increasing or decreasing). This proves the tightness.