\documentclass[11pt]{article}
\usepackage{fullpage}
\usepackage{url}
\usepackage{amsmath}

% This is a file of macros and definitions that may come up in
% ANY LATEX paper.  Typically I'll use this file and some other file in the
% directory of the paper, this one for general math things, that one for
% things specific to that paper.
%
% font's used and general paper things.
\font\tenrm=cmr10
\font\ninerm=cmr9
\font\eightrm=cmr8
\font\sevenrm=cmr7
% \font\title=cmbx10 scaled \magstep1 % extra big title font
\font\ss=cmss10 % used by \proof
\font\smallcaps=cmcsc10 % used to label Theorems, etc.
% imhibit black bars on overflows
%
\overfullrule=0pt
%
% today's date
%
%
% English words that I always italizice in papers.
% Some words that appear in math mode alot that I wasn roman
%

\newcommand{\PVDW}{{\rm PVDW}}
\newcommand{\AREA}{{\rm AREA}}
\newcommand{\ALG}{{\rm ALG}}
\newcommand{\ACK}{{\rm ACK}}
\newcommand{\VC}{{\rm VC}}
\newcommand{\AC}{{\rm AC}}
\newcommand{\ZL}{{\rm ZL}}
\newcommand{\WOP}{{\rm WOP}}
\newcommand{\tree}{{\rm tree}}
\newcommand{\SUBSEQ}{{\rm SUBSEQ}}
\newcommand{\minor}{{\preceq_m}}
\newcommand{\minorp}{{\preceq_m'}}
\newcommand{\cminor}{{\preceq_{\rm c\hbox{-}m}}}
\newcommand{\GM}{{\rm  GM}}
\newcommand{\TREE}{{\rm  TREE}}
\newcommand{\R}{\rm  R} 
\newcommand{\B}{\rm  B} 
\newcommand{\G}{\rm  G} 
\newcommand{\Y}{\rm  P} 

\newcommand{\IO}{\exists^\infty}
\newcommand{\ceil}[1]{\left\lceil {#1}\right\rceil}
\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}

\newcommand{\into}{{\rightarrow}}
\newcommand{\D}{{\sf D}}
\newcommand{\N}{{\sf N}}
\newcommand{\Z}{{\sf Z}}
\newcommand{\Q}{{\sf Q}}
\newcommand{\reals}{{\sf R}}
\newcommand{\qd}{{\rm qd}}
\newcommand{\GE}[1]{{\equiv^{G}_{#1}}}
\newcommand{\TE}[1]{{\equiv^{T}_{#1}}}
\newcommand{\TES}[1]{{\equiv^{T}_{#1,{\rm S}}}}
\newcommand{\TEB}[1]{{\equiv^{T}_{#1,{\rm BOOL(S)}}}}
\newcommand{\LL}{{\cal{L}}}
\newcommand{\onepe}{1+\epsilon}
\newcommand{\oneme}{1-\epsilon}
\newcommand{\cool}{cool}
\newcommand{\nze}{numzeros}
\newcommand{\non}{numones}
\newcommand{\LR}[2]{ {\rm LR}_{{#2}}({#1})}
\newcommand{\LH}[2]{ {\rm LH}({#1};{#2})}
\newcommand{\bbb}{{\gamma}}
\newcommand{\COL}{{\rm COL}}
\newcommand{\card}[1]{\#(#1)}
\newcommand{\st}{\mathrel{:}}
\newcommand{\fhat}{{\hat{f}}}
\newcommand{\Ahat}{{\hat{A}}}

\begin{document}

\centerline{\bf AT-Home Question for Final, And An Extra Credit Question}
\centerline{\bf Morally Due TUESDAY May 12}

Note: Dead-Cat is THUSDAY MAY 14.

Our final is FRIDAY May 15.

You are WELL ADVISED to do this SOON so that its out of the way
while you are studying for your finals. 







\begin{enumerate}

\item
(0 points) What is your name? 


\vfill

\centerline{\bf GO TO NEXT PAGE}

\newpage

\item
(20 points of the Final) 

Prove the following:

{\it For all $c$, For all $\COL \colon \reals^2 \into [c]$ there exists three points
that are the same color such that the area of the triangle they form is 1.} 


\vfill

\centerline{\bf GO TO NEXT PAGE}

\newpage

\item 
(0 points but this is your last extra credit problem, so your last change to impress me.)

Assume that you already have proven $\PVDW(\omega,\omega)$. 

Prove the following:

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

$\COL(a) = \COL(a+d^3)$.

}

(If you have questions for me about this problem, FIRST read through and UNDERSTAND
the slides on Poly VDW.) 

\end{enumerate}

\end{document}

