\documentclass[11pt]{scrartcl}
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\begin{document}
\title{Shortlist 1999 C6}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 77}
\maketitle
\section*{Problem}
Suppose that every integer has been given one of the colours red, blue,
green or yellow. Let $x$ and $y$ be odd integers so that $|x| \neq |y|$.
Show that there are two integers of the same colour whose difference has
one of the following values: $x,y,x+y$ or $x-y$.
\section*{Video}
\href{https://www.youtube.com/watch?v=iznvJAYuUqo&list=PLi6h8GM1FA6yHh4gDk_ZYezmncU1EJUmZ&index=19}{\texttt{https://youtu.be/iznvJAYuUqo}}
\newpage
\section*{Solution}
Assume for contradiction a coloring
\[ \chi \colon \ZZ \to S \defeq
\left\{ \text{red}, \text{green}, \text{blue}, \text{yellow} \right\}
\]
existed violating the conclusion.
Then, we construct a coloring of $\wh{\chi} \colon \ZZ^2 \to S$ by
\[ \wh{\chi} \left( a,b \right) = \chi(x \cdot a + y \cdot b). \]
\begin{claim*}
$\wh{\chi}$ assigns different colors to $(a,b)$, $(a,b+1)$, $(a+1,b)$, $(a+1,b+1)$.
\end{claim*}
\begin{proof}
By definition.
\end{proof}
However colorings $\wh{\chi} \colon \ZZ^2 \to S$ satisfying the claim
are actually straightforward to describe completely.
Once such a description is given,
one can directly check it can't be the lift of a $\chi$ as described.
\end{document}