\documentclass[12pt,ifthen]{article}
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\begin{document}

\centerline{\bf CMSC 752 Homework 12} 
\centerline{\bf Morally Due Tue April 29, 2025} 
\centerline{\bf Dead Cat May 1}

\bigskip

\newif{\ifshowsoln}
\showsolntrue % comment out to NOT show solution inside \ifshowsoln and \fi blocks.

\smallskip

\begin{enumerate} 

\item 
(50 points) 
Let $k\in\N$. Show the following

{\it $\forall c$ $\exists W=W(c)$ such that, $\forall$ $\COL\colon [W]\into [c]$
$\exists a,d$ such that 

$$a, a+d^2+d, a+d^2+2d, \ldots, a+d^2+kd$$

are all the same color.}

You may assume VDW's theorem, which  the cool kids call, $\PVDW(\omega)$. 

\newpage

\item
(50 points)
Show the following

{\it $\forall c$ $\exists W=W(c)$ such that, $\forall$ $\COL\colon [W]\into [c]$
$\exists a,d$ such that 

$$a, a+d^3$$

are all the same color.}

You may assume $\PVDW(\omega,\omega)$. 


\end{enumerate}

\end{document}