Two-colour Schur theorem

Every red-blue colouring of the positive integers contains a monochromatic triple $x,y,z$ with $x+y=z$; the summands need not be distinct. Colour each edge $ij$ of $K_6$ by the colour of $|i-j|$. Since the Ramsey number $R(3,3)$ is at most $6$, there is a monochromatic triangle $i<j<k$. Its positive edge differences $j-i$, $k-j$ and $k-i$ have one colour and satisfy the required sum equation.