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I want to find an example of a sequence of n=rs distinct numbers where there is not an increasing sequence of length s or a decreasing sequence of length r, hence showing that the bound given by Erdős-Szekeres is the best possible.

However, having tried several options, I cannot come up with a single example. I thought of using r=1=s, but this gives a single element sequence which is both increasing and decreasing so does not work.

I am starting to think it cannot be possible. Is there such an example?

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  • $\begingroup$ See here, here, or here for the case $r=s$; the idea is readily generalized. $\endgroup$ – Brian M. Scott Feb 11 '15 at 22:38
  • $\begingroup$ You are misstating the Erdős-Szekeres result. It says that, in a sequence of $rs+1$ distinct numbers, there is either a decreasing subsequence of length $r+1$ or an increasing subsequence of length $s+1$. This is best possible because you can construct a sequence of length $rs$ with no decreasing subsequence of length $r+1$ and no increasing subsequence of length $s+1$. $\endgroup$ – bof Feb 11 '15 at 22:48
  • $\begingroup$ I think there was a typo in the original result I was given. Makes sense I wasn't able to disprove it if what I was actually doing was within the bounds of the theorem! $\endgroup$ – AccioHogwarts Feb 24 '15 at 4:51
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Take the sequence $\color{red}{r,r-1,r-2\dots 1},\color{blue}{2r,2r-1,2r-2\dots r+1} \dots \color{green}{rs,rs-1\dots (s-1)r+1} $

If you want an increasing sequence you can't take two of the same color, if you want a decreasing sequence you can't take two of different colors.

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