Looking for references concerning combinatorial questions

Hello,

I am looking for references concerning something. I think it has some
relations with combinatorial mathematics, but I can't find the exact
name of the thing. Let's assume you have an infinite sequence of elements
taken in a finit set ; for instance the digits of an irrational number:
.141592653...
I am looking for theorems that would answer to questions like:
a) is there a periodical subsequence of n terms ?
b) is there a symetrical subsequences of n terms ?
c) how many initial terms should we take in order to be sure
we have a subsequence with such or such pattern of length n in it ?
etc.

As you can see, I am interested in local "patterns" that may be found
in this infinite subsequece (whith elements taken in a finite set).

What is the exact name of the part of mathematics I am speaking of ?
Where can I find (web preferred) theorems as described above ?

Cordially,

--
Lord Jim
.