Can you arrange the numbers from 1 to 100 so there is no ascending or descending subsequence of length greater than 10?

The items in a subsequence need not be adjacent. For example, (2, 5) is an ascending subsequence of (1,4,2,6,3,5,7), though not the longest one.

Hint: What is the relationship between 10 and 100? Can you create a simpler problem using smaller numbers and apply the solution to this problem?

There is a lot of mathematics around generalizations of this problem. I will give a followup problem based on a really slick proof that the numbers from 1 to 101 must contain either an ascending or descending sequence of length at least 11.