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.



Erdos-Szekeres Theorems
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](http://www.ams.org/journals/bull/2000-37-04/S0273-0979-00-00877-6/home.html).


