The numbers $1,2,3,...,101$ are written down in a row in some order. Is it always possible to cross out $90$ numbers in a way such that all $11$ numbers left will stay either in increasing or in decreasing order?
2026-03-26 17:51:32.1774547492
Interesting question with ordering integers
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The answer is yes, and this is due to a result of Erdos and Szekeres. A particular case of their theorem states that : for every sequence of $n^2+1$ real numbers, there is either an increasing or decreasing subsequence of length $n+1$.
The proof is either by Pigeonhole Principle or Dilworth's Theorem and can be found here: http://en.wikipedia.org/wiki/Erdős–Szekeres_theorem