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I'm following a Combinatorics course at the moment, and have recent proved the Erdős–Szekeres Theorem (or, at least, some variation of):

A sequence of length $n^2 + 1$ either contains an increasing subsequence of length $n+1$, or a decreasing one of length $n+1$.

I have been set an exercise to prove the following:

For each natural number $n$, find a sequence of $n^2$ real numbers which contains no monotonic subsequence of more than $n$ terms.

I can naively / informally how to do this, but no method which would seem to involve any sort of Graph / Ramsey Theory. I'm assuming that I'll have to use Erdős–Szekeres, since I'm in a Combinatorics course, or notice something about its proof, to come up with a formal construction of the sequence, but can't see how I'd go about it.

Would using Erdős–Szekeres be the right way to go about this, or is there a different / simpler combinatorial way to explore?

Thanks!

Martin Sleziak
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Mystery_Jay
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1 Answers1

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Hint: $$\def\b{\color{red}{\bullet}\ }\def\w{\circ\ } \eqalign{\w\ \w\w\w\w\w\b\w\w\cr \w\ \w\w\w\w\w\w\b\w\cr \w\ \w\w\w\w\w\w\w\b\cr \w\ \w\w\b\w\w\w\w\w\cr \w\ \w\w\w\b\w\w\w\w\cr \w\ \w\w\w\w\b\w\w\w\cr \b\ \w\w\w\w\w\w\w\w\cr \w\ \b\w\w\w\w\w\w\w\cr \w\ \w\b\w\w\w\w\w\w\cr}$$

David
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  • Yes, this was the 'informal' idea that I had; glad to see what I was thinking was indeed on the right lines.

    I've thought about it some more and been able to generalise it a bit more, so thanks for confirming my thoughts!

     – Mystery_Jay Nov 27 '14 at 19:15