\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}