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Topic: Pigeonhole Principle ?
Replies: 9   Last Post: Oct 14, 2013 2:15 AM

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Virgil

Posts: 9,012
Registered: 1/6/11
Re: Pigeonhole Principle ?
Posted: Oct 8, 2013 7:33 PM
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In article <25f8c9b1-2a29-434c-88d0-421736612be6@googlegroups.com>,
grahamcooper7@gmail.com wrote:

> On Sunday, October 6, 2013 7:55:59 PM UTC-7, shaoyi he wrote:
> > 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.

How about
Every sequence of 2*n + 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?

>
>
> Every list MUST have some increasing/decreasing sublist (although much
> smaller).
>
> It seems trivially false.


If the n^2 + 1 case is false then so must be the 2*n+1 case,
since n^2 + 1 > 2*n+1 for n > 1.

So can anyone find a list of 2*n+1 different integers that does NOT have
either a sublist of n increasing integers or a sublist of n decreasing
integers?
--





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