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\begin{document}

\centerline{\bf CMSC 752 Homework 13} 
\centerline{\bf Morally Due Tue May 6, 2025} 
\centerline{\bf Dead Cat May 8}

\bigskip

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\begin{enumerate} 

\item
(50 points)
For this problem you can assume the Gallai-Witt Theorem in two dimensions which we
state here for your convinence.

{\it For all $k$, for all $c$, there exists $GW=GW(k,c)$ such that for all 

$$\COL\colon [GW] \times [GW] \into [c]$$

there exists a monochromatic $k\times k$ equally spaced grid. More preciesly
there exists $a,b,d)$ such that the following are all the same color:

$$\{(a+id,b+jd)\colon 0\le i,j\le k-1\}.$$
}

Prove the following which is the Can VDW theorem.

{\it For all $k$ there exists $C=C(k)$ such that, for all $\COL \colon [C]\into [\omega]$
one of the two occurs
\begin{itemize}
\item
There exists $a,d$ such that $a, a+d, \ldots, a+(k-1)d$ are the same color.
\item
There exists $a,d$ such that $a, a+d, \ldots, a+(k-1)d$ are all different colors. 
\end{itemize}
}

{\bf Hint} given $\COL\colon [W]\into [\omega]$ form 

$\COL'\colon [W']\times [W']\into [c]$ ($W'$ and $c'$ to be deterined by you)
as follows:


$\COL'(a,d)$ is determined by looking at 

$X=\{\COL(a), \COL(a+d),\ldots,\COL(a+kd)\}$

and outputing the $k+1$-tuple

Number of times $\COL(a)$ appears in  $X$.

Number of times $\COL(a+d)$ appears in  $X$.

$\vdots\qquad\vdots\qquad\vdots$

Number of times $\COL(a+kd)$ appears in  $X$.

For example, if the  

$\COL(a)=R$

$\COL(a+d)=R$

$\COL(a+2d)=B$

$\COL(a+3d)=Y$

$\COL(a+4d)=R$

$\COL(a+5d)=B$

Then 

$\COL'(a,d)=(3,3,2,1,3,2)$

\newpage

\item 
(50 points) Use the Prob Method to get a lower bound on $W(k,c)$.
(This DOES NOT use the fancy Lovasz Local Lemma.) 

\end{enumerate}

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