Correction: In the last sentence, one should change one of my original increasings to decreasing. It doesn't matter which one, of course: << This question sounded familiar, and in fact it's a special case or the Erdos-Szekeres theorem (1935): < http://en.wikipedia.org/wiki/Erdős–Szekeres_theorem >, original paper at < http://archive.numdam.org/ARCHIVE/CM/CM_1935__2_/CM_1935__2__463_0/CM_1935__... >. The theorem states that any set of integers of size (r-1)(s-1) + 1 has either an increasing subsequence of length r or an decreasing subsequence of length s.
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