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# Monotonic Subsequences

## Question

Consider a finite sequence of distinct integers. A subsequence is a sequence formed by deleting some items from the original sequence without disturbing their relative ordering. A subsequence is called monotone if it is either increasing (each term is larger than the one before it) or decreasing (each term is smaller than the one before it). For example, if the sequence is 4, 6, 3, 5, 7, 1, 2, 9, 8, 10, then 4, 6, 8, 10 is a monotone (increasing) subsequence of length 4 and 6, 5, 2 is a monotone (decreasing) subsequence of length 3.
a) Find a sequence of 9 distinct integers that has no monotone subsequence of length 4.
b) Show that every such sequence of length 10 has a monotone subsequence of length 4.
c) Generalize. How long must the sequence be to guarantee a monotone subsequence of length n?

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a) Nine distinct integers with no monotone sub-sequence of length 4.

7 8 9 4 5 6 1 2 3

which can be visualized better as follows:

9

8

7

6

5

4

3

2

1

Every number has at most two larger that follow it in sequence.

Every number has at most two smaller that follow it in sequence.

b) The structure suggests there is no place to insert 0, say, without disrupting this property.

That is, this structure (or equivalent permutation) is needed to avoid 4, and inserting another number destroys it.

c) By extension of (1 + 32) to force length 4, it seems likely to take (1 + (n-1)2) numbers to force length n.

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