In article <email@example.com>, firstname.lastname@example.org 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? --