A monotone subsequence is a sequence derived from a given sequence where the elements are either entirely non-increasing or non-decreasing. This means that within the subsequence, each element is either greater than or equal to (in the case of a non-decreasing subsequence) or less than or equal to (for a non-increasing subsequence) the preceding element. Monotone subsequences are key in understanding certain properties of sequences, especially in the context of the Erdős-Szekeres Theorem, which addresses the existence of such subsequences in larger sequences.
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