Ramsey's Number Theorem Proof

Proof:

w->(w)ⁿ_k
is proven by induction of 'n'.

If n=1, then it just states that any partition of an infinite set into a finite number of subsets must //include an infinite set; (i.e.,the union of a finite number of finite sets is finite).

This would be easier to prove, as there are finite number of sets, there is the largest set of size x. Let the number of sets be 'y'. Then the size of the union is not more than xy.

If w->(w)ⁿ_k
then it can be represented as, w->(w)ⁿ⁺¹_k

Let 'f' be some coloring of [S]ⁿ⁺¹ by k, Where 'S' is an infinite subset of 'w'.

The value of x can be defined as:
fx:[Sx]ⁿ->k by f^x(X)=f(x U X).

As 'S' is infinite, by the induction hypothesis, this is said to have an infinite homogeneous set.

Sequence of Integers (ni)iw and Sequence of Infinite Subsets of w , (Si)iw can be defined by induction.

Let n₀=0 and let S₀=w.
Given n_i and S_i for i<=j we can define S_j as an infinite homogeneous set for fⁿⁱ:[S_j−1]ⁿ->k and n_jas the least element of S_j . Obviously N=n_i is infinite, and it is also homogeneous, since each n_i is contained in S_j for each j_i .