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\newcommand{\VDW}{{\rm VDW}}
\newcommand{\PVDW}{{\rm PVDW}}
\newcommand{\degg}{{\rm deg}}
\newcommand{\spec}{{\rm spec}}
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\newcommand{\COL}{{\rm COL}}
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\newcommand{\PF}{{P^{finite}}}
\newcommand{\ceil}[1]{\lceil #1 \rceil}

\usepackage{amsthm}
\newtheorem{theorem}{Theorem}



\begin{document}


\centerline{\bf Homework 12} 

\centerline{\bf Morally Due Tue May 3 at 3:30PM. Dead Cat May 5 at 3:30}

\centerline{\bf WARNING: THE HW IS TWO PAGES LONG}

\begin{enumerate}

\item
(0 points)
What is your name?  Write it clearly.
When is the take-home final due?

\item
(50 points) 
Let $VDW(k,c)$ be the statement

{\it There exists $W=W(k,c)$ such that for all $\COL\colon [W]\into[c]$ there exists $a,d\ge 1$ such that

$$a, a+d, \ldots, a+(k-1)d \hbox{ are the same color }.$$

}

Let $W(k,c)$ be as in the statement. 

AND NOW FOR THE PROBLEM

Assume $(\forall c)[VDW(9,c)]$. Prove $\VDW(10,2)$. Your proof should give an upper bound on $W(10,2)$ 
as a function of $W(9,c)$.



\vfill

\centerline{\bf GO TO NEXT PAGE}

\newpage

\item 
(50 points)
In this problem you will do PART of the proof of 

$(\forall k)[\PVDW(x^2,x^2+x,\ldots,x^2+kx)]$. 


\begin{enumerate}
\item
(20 points) 
State carefully the LEMMA that will imply 

$(\forall k)[\PVDW(x^2,x^2+x,\ldots,x^2+kx)]$. 

\item 
(30 points) 
Prove carefully the BASE CASE of that lemma. 

\item
(0 points, but good for your enlightenment)
Prove the Induction Step of the lemma. 
\end{enumerate}

\vfill

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\newpage







\end{enumerate}



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