\documentclass[12pt,ifthen]{article}
\usepackage{amsmath}
\usepackage{amssymb}
%\usepackage{html}
\usepackage{hyperref}
\newcommand{\degg}{{\rm deg}}
\newcommand{\spec}{{\rm spec}}
\newcommand{\PH}{{\rm PH}}
\newcommand{\COL}{{\rm COL}}
\renewcommand{\S}{{\sf S}}
\newcommand{\N}{{\sf N}}
\newcommand{\Z}{{\sf Z}}
\newcommand{\R}{{\sf R}}
\newcommand{\into}{{\rightarrow}}
\newcommand{\PF}{{P^{finite}}}
\newcommand{\ceil}[1]{\lceil #1 \rceil}

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


\begin{document}


\centerline{\bf Homework 11} 

\centerline{\bf Morally Due Tue April 26 at 3:30PM. Dead Cat April 28 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) 
Find a value of $m<50$ such that the following holds, and prove it.
Prove it from first principles--- that is, your proof should not refer to
the slides or any other source.

{\it For all 3-colorings of the $4\times m$ grid, there exists a mono rectangle.}


\vfill

\centerline{\bf GO TO NEXT PAGE}

\newpage

\item 
(50 points)
For this problem we can assume the following is known:

{\it For all $c\ge 1$ there exists $L=L(c)$ such that 

for all $c$-colorings of the $L(c)\times L(c)$ grid 

there exists a monochromatic isocles $L$.}

Let a big-base-$L$ be the same shape as the following four points:

$(0,0)$, $(d,0)$, $(2d,0)$, and $(0,d)$. 

And NOW for our problem:


{\it Show that there exists $LL$ such that, 

for all 2-colorings of the $LL\times LL$ grid 

there exists a monochromatic big-base-$L$.}

Feel free to use PICTURES in your proof. 


\end{enumerate}

\end{document}