\documentclass[12pt,ifthen]{article} \usepackage{amsmath} \usepackage{amssymb} %\usepackage{html} \usepackage{hyperref} \newcommand{\degg}{{\rm deg}} \newcommand{\PVDW}{{\rm PVDW}} \newcommand{\VDW}{{\rm VDW}} \newcommand{\COL}{{\rm COL}} \renewcommand{\S}{{\sf S}} \newcommand{\Q}{{\sf Q}} \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 Final} \centerline{\bf Morally DUE Monday May 16 at 3:30PM. Dead Cat Wed May 18 at 3:30} \begin{enumerate} \item (0 points) What is your name? Write it clearly. \item (24 points) Let $R_a(k)$ be the least $n$ such that \centerline{\it for all $\COL\colon\binom{[n]}{a}\into[2]$ there exists a homog set of size $k$.} Assume that Zan and Not-Zan have shown that $R_3(k)\le 2^{100k}$. Using this to find an upper bound on $R_4(k)$ of the form $R_4(k)\le 2^{2^{dk}}$. Give the $d$ and the proof. \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (24 points) Prove or Disprove: {\it For every $\COL\colon \Q \into [\omega]$ there exists an $H\subseteq \Q$ such that \begin{itemize} \item $H$ has the same order type as the rationals which means all of the following hold: a) $H$ is countable b) $H$ is dense: \$(\forall x,y\in H)[x