Erdos-Szekeres Theorems 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.
 A: They are not the same. Erdős and Szekeres (when they were students) proved that for any $k$ there is $n$ such that among any $n$ points in the plane (in general position) there are $k$ points which form a convex polygon. In the course of the proof they rediscovered Ramsey's theorem (the one that you quote about complete graphs or perhaps a more general version for hypergraphs). Erdős and Szekeres were not aware of Ramsey's work. See also this survey.
A: Firstly, I second Qiaochu's remark that I've never heard the Ramsey theorem referred to as Erdős-Szekeres' theorem. 
Secondly, it is is true (and actually well-known) that the Ramsey theorem implies a kind of a "weak version" of the Erdős-Szekeres theorem. Namely, given an $n$-term sequence $\{a_1,...,a_n\}$, consider the complete graph on the vertex set $[n]$, coloring the edge $(i,j)$ with $1\le i<j\le n$ blue if $a_i\le a_j$, and red if $a_i>a_j$. Now if $n>R(s,t)$, then our graph has either blue complete subgraph on $s$ vertices, corresponding to a length-$s$ increasing subsequence, or a red complete subgraph on $t$ vertices, corresponding to a length-$t$ decreasing subsequence.
