```Date: Oct 7, 2013 2:31 AM
Author: Jussi Piitulainen
Subject: Re: Pigeonhole Principle ?

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 + 1distinct numbers) so that there is no increasing or decreasingsubsequence 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 allowa subsequence like 3 4 6 7 where the numbers are not adjacent in theoriginal sequence.
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