Here's the question:
Describe a sequence consisting from 1 to 10,000 in some order so that there is no increasing or decreasing subsequence of size 101.
I'm not quite sure how to do this. My first thought was to put the first 100 in increasing order, then the next 100 in decreasing order, but I dismissed that because you could make a large subsequence of that, I think.
My next thought was probably more on the right track, that 10,000 is 100 * 100, so it must have something to do with that. I got no farther with that. Any help would be greatly appreciated.
Edit: If it helps to get a scope of what I've learned thus far, the part that this question is being answered in is about graphs, relations, partial orders, chains, sequences, etc.
Your first thought works: start with $$100,99,98,\ldots,3,2,1\;,$$ continue with $$200,199,198,\ldots,103,102,101\;,$$ and so on. Any subsequence of $101$ numbers must have two numbers in the same block of $100$ and two numbers in different blocks; the first two are decreasing, and the second two are increasing, so the subsequence as a whole is not monotone.