proof of Ramsey’s theorem
is proven by induction on .
If then this just states that any partition of an infinite set into a finite number of subsets must include an infinite set; that is, the union of a finite number of finite sets is finite. This is simple enough to prove: since there are a finite number of sets, there is a largest set of size . Let the number of sets be . Then the size of the union is no more than .
then we can show that
Then we define a sequence of integers and a sequence of infinite subsets of , by induction. Let and let . Given and for we can define as an infinite homogeneous set for and as the least element of .
|Title||proof of Ramsey’s theorem|
|Date of creation||2013-03-22 12:55:52|
|Last modified on||2013-03-22 12:55:52|
|Last modified by||mathcam (2727)|