> [!NOTE] Lemma > There does not exist a function $f:\mathbb{Z}\to \{ 1,2,3 \}$ such that for all $x,y\in \mathbb{Z}$ with $|x-y|\in \{ 2,3,5 \}$ $f(x)\neq f(y).$ **Proof**: BWOC, suppose there exists such a function $f.$ WLOG, let $f(0)=1$ and $f(5)=2.$ Since $|5-2|=3, |2-0|=2,$ we have $f(2) = 3.$ Also since $|5-3|=2,|3-0|=3,$ we must have $f(3)=f(2).$ Repeating the argument with $x$ instead of zero, we gain $f(x+2)=f(x+3)$ so $f$ is constant contradicting the initial assumption. # Application Takeaway: we made the choice of starting with $f(0)$ for the first argument. See want happens if we make an arbitrary choice.