Why is Erdős–Szekeres theorem sharp?
Solution 1:
For $n=5$:
$$\underbrace{5,4,3,2,1},\underbrace{10,9,8,7,6},\underbrace{15,14,13,12,11},\underbrace{20,19,18,17,17},\underbrace{25,24,23,22,21}$$
Any subsequence of $6$ of these numbers must contain two numbers from the same block and two numbers from different blocks. Two numbers from the same block must appear in decreasing order, while two from different blocks must appear in increasing order. Thus, the subsequence cannot be monotonic.
The construction clearly generalizes to arbitrary $n$.