Schur's Theorem (Ramsey Theory)

Theorem
For every positive integer $$r$$, there exists a positive integer $$S$$, such that for every partition of the integers $$\left\{{1, \ldots, S}\right\}$$ into $$r$$ parts, one of the parts contains integers $$x$$, $$y$$ and $$z$$ such that:
 * $$x + y = z$$.

Proof
Let $$n = R \left({3, \ldots, 3}\right)$$ where $$R \left({3, \ldots, 3}\right)$$ denotes the Ramsey number on $$r$$ colors.

Now take $$S$$ to be $$n$$ and partition the integers $$\left\{{1, \ldots, n}\right\}$$ into $$r$$ parts, which we denote by colors.

That is: ... and so on till color $$c_r$$.
 * The integers in the first part are said to be colored $$c_1$$;
 * The integers in the second part are said to be colored $$c_2$$

We also then say that $$\left\{{1, \ldots, S}\right\}$$ has been $$r$$-colored. This terminology is common in Ramsey theory.

Now consider the complete graph $$K_n$$.

Now color the edges of $$K_n$$ as follows:


 * An edge $$xy$$ is given color $$c$$ if $$\left|{x - y}\right|$$ was colored $$c$$ in the partitioning.

Now from the definition of $$R \left({3, \ldots, 3}\right)$$ and Ramsey's Theorem, $$K_n$$ will definitely contain a monochromatic triangle, say built out of the vertices $$i > j > k$$.

Suppose the triangle is colored $$c_m$$. Now $$i - j$$, $$i - k$$ and $$j - k$$ will also be colored $$c_m$$, i.e. will belong to the same part in the partition.

It only remains to take $$x = i - j$$, $$y = j - k$$ and $$z = i - k$$ to complete the proof.