Example of Erdos-Szekeres bound being tight

Question

I know that Erdos-Szekeres Theorem states that any sequence of $k^2+1$ real numbers contains a monotonic subsequence of length at least $k+1$. I know that it is a least upper bound since there exist sequences of $k^2$ real numbers that do not contain a monotonic subsequence of length $k+1$.

First, I was asked to prove the Erdos-Szekeres Theorem in two different ways, which was done accordingly. Then, I was asked to Describe an example which shows that the bound given by the Erd}os-SzekeresTheorem is tight.

I have little clue on how to approach this. Any suggestions?

Answer 1

HINT: $k=4$:

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