I found two theorems called "Erdos-Szekeres" theorem, I am not sure they're the same. The first one is about ordered sequences of numbers:
For any sequence of (r-1)(s-1)+1 distinct numbers, there is either an increasing sequence of length r or a descreasing subsequence of length s
Here's one about complete graphs:
Given a pair of integers s,t there is an integer, R(s,t) such that any 2-coloring of complete graph on n vertices has a red complete graph on s vertices or a blue complete graph on t vertices.
I have seen this in other contexts, a ramsey theory problem might be graph-theoretic in one version and combinatorial or number-theoretic in the other.