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\centerline{\bf Homework 9, Morally Due Tue May 5, 2020, 3:30PM}

COURSE WEBSITE: \url{http://www.cs.umd.edu/~gasarch/858/S18.html}

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\centerline{\bf THIS HW IS ONE PAGES LONG!!!!!!!!!!!!}

\bigskip

\begin{enumerate}
\item
(0 points)
What is your name?  Write it clearly.

\item
(50 points)
In this problem you may assume that, for all $c$, there exists $N=N(c)$
such that for all $c$-colorings of $[N]\times[N]$ there exists a 
monochromatic square.

Show that there exists $M$ such that, for all 2-colorings of $[M]\times[M]$,
there exists five points that are the same color of the following form:

$(x,y)$

$(x+d,y)$

$(x,y+d)$

$(x+d,y+d)$

$(x+2d,y+d)$

(This is called a {\it Little Dipper}.)


You can (and should) prove this by making drawings and pointing to stuff.

\item (50 points) 
Assume that you know that, for all $c$, $W(100,c)$ exists.
Prove that $W(101,2)$ exists. 
You can draw diagrams; however, your proof should be completely rigorous.




\end{enumerate}

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