The Erdős-Szekeres theorem states that if , , and , ..., is a sequence of real numbers, then this sequence contains a monotonic increasing (decreasing) subsequence of terms or a monotonic decreasing (increasing) subsequence of terms.
Dilworth's lemma is a generalization of this theorem.
More things to try:
Weisstein, Eric W. "Erdős-Szekeres Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Erdos-SzekeresTheorem.html
, 
, and 
, ..., 
is a sequence of 
real numbers, then this sequence
contains a monotonic increasing (decreasing)
subsequence of 
terms or a monotonic decreasing (increasing)
subsequence of 
terms.
