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Re: Pigeonhole Principle ?
Posted:
Oct 7, 2013 2:31 AM


shaoyi he <wubuqilai@gmail.com> writes:
> in discrete mathematics and its applications 6th ,in Pigeonhole > Principle, the author give a THEOREM(page 351): > > Every sequence of n^2 + 1 distinct real numbers contains a > subsequence of length n + 1 that is either strictly > increasing or strictly decreasing. > > i donnot know what's the theorem for? because when we sort the n^2 + > 1 distinct real numbers, we can get n^2+1 that is either strictly > increasing or strictly decreasing. so how to understand this?
It's saying there is no way to arrange 0 1 2 3 4 5 6 7 8 9 (3^2 + 1 distinct numbers) so that there is no increasing or decreasing subsequence of 4 (3 + 1) numbers.
("Strictly" is redundant when the numbers are distinct.)
Checking something:
In 3 1 4 5 9 2 6 8 7 0, there's 1 4 5 9.
In 3 1 5 4 9 2 6 8 7 0, there is no such, so the statement must allow a subsequence like 3 4 6 7 where the numbers are not adjacent in the original sequence.



