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 + 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.