\documentclass[12pt,ifthen]{article}
\usepackage{amsmath}
\usepackage{hyperref}
\usepackage{comment}
\newcommand{\N}{{\sf N}}
\newcommand{\Q}{{\sf Q}}
\newcommand{\Z}{{\sf Z}}
\newcommand{\into}{{\rightarrow}}
\newcommand{\st}{\mathrel{:}}
\newcommand{\IO}{\exists^\infty}
\newcommand{\COL}{\mathrm{COL}}
\newcommand{\lf}{\left\lfloor}
\newcommand{\rf}{\right\rfloor}
\newcommand{\lc}{\left\lceil}
\newcommand{\rc}{\right\rceil}
\newcommand{\Ceil}[1]{\left\lceil {#1}\right\rceil}
\newcommand{\ceil}[1]{\left\lceil {#1}\right\rceil}
\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}
\newcommand{\nth}{n^{th}}
\begin{document}
\centerline{\bf Take Home Final}
\centerline{\bf Due May 20 at 7:00AM. No Extensions}
\newif{\ifshowsoln}
% \showsolntrue % comment out to NOT show solution inside \ifshowsoln and \fi blocks.
\centerline{\bf THIS EXAM IS THREE PAGES!!!!!!!!!!!!!!!}
\begin{enumerate}
\item
(0 points)
What is your name? Write it clearly.
\item (20 points) % Ramsey
A coloring $COL:\binom{\N}{2}\into \N$ is {\it Dhruv} if, for all $x\min(H)$ (note the strict inequality).
In class we showed $LR_2(2)\le 13$ and there are now slides on that on the course websites.
Prove that $LR_2(2)\le 12$.
\item
(20 points) % VDW
For this problem you MAY NOT assume Rado's theorem. You will essentially be proving
it in a particular case.
Fill in an $A\ge 1$ and a $B\ge 1$ such that the following 2-part theorem is true, and prove both parts.
\begin{enumerate}
\item
There exists a $c$ and a $c$-coloring of $\N$ such that the equation
$$w+2x+5y-Az=0$$
has NO mono solution.
Try to make $A$ as small as possible.
\item
For every $c$ there exists $n$ such that for all $c$-colorings of $[n]$
there is a mono solution to the equation
$$w+2x+5y-Bz=0$$
with $w,x,y,z$ ALL DIFFERENT.
Try to make $B$ as large as possible.
(YES I know that for this part I said ALL DIFFERENT and for Part a I did not.
This is NOT a mistake.)
\end{enumerate}
\end{enumerate}
\end{document}
\end{enumerate}
\end{document}