Arrange the integers 1-100 in such a way that no eleven of the numbers selected from left to right (adjacent or otherwise) form a strictly increasing or decreasing sequence. (This means if 11 numbers are selected from anywhere in the pattern they cannot be in an increasing or decreasing pattern).
I think I have to divide the 100 integers into 10 groups, all with 10 integers. But that's all I've thought about.
On
I'd like to point out another construction: $$91,81,71,\ldots,92,82,72,\ldots,93,83,73,\ldots $$ If you want an increasing sequence, no matter which number you choose you must pass at each step to the next block of ten numbers $\implies$ at most ten choices.
If you want a decreasing sequence you must lower at each step the ten's figure $\implies$ at most ten choices.
To reiterate my comment from above and remove this from the unanswered queue:
Think about the smaller question of arranging the numbers $1$-$9$ in such a way that there is no monotonic subsequence of length $4$ or more.
$789~456~123$ satisfies this condition. If you were to create an increasing subsequence all of the numbers must be from the same block. If you were to create a decreasing subsequence all of the numbers must be from different blocks. As such, the largest length increasing subsequence is the size of a block and the largest length decreasing subsequence is the number of blocks, both of which in this case are three.
Generalize this to your specific problem.