 Solution

Colour the edge $ij$ of the complete graph on $\{1,\dots ,6\}$ by the class containing $|j-i|$. Part (a) gives $R(3,3)\le 6$, so there is a monochromatic triangle with vertices $i<j<k$. Its three edge differences areThey are positive integers in one class. Hence every two-colouring of the positive integers has a monochromatic solution of $x+y=z$. This is the two-colour Schur theorem. The two summands need not be distinct.