= Solution
Colour the <edge> $ij$ of the <complete graph> on $\{1,\ldots,6\}$ by the class containing $|j-i|$. Part (a) gives $R(3,3)\leq6$, so there is a monochromatic triangle with vertices $i<j<k$. Its three <edge> differences are
$$
x=j-i,\qquad y=k-j,\qquad z=k-i=x+y.
$$
They are positive integers in one class. Hence \b[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.
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