\documentclass[12pt,ifthen]{article} \usepackage{amsmath} \usepackage{amssymb} \usepackage{comment} %\usepackage{html} \usepackage{hyperref} \newcommand{\VDW}{{\rm VDW}} \newcommand{\PVDW}{{\rm PVDW}} \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 12} \centerline{\bf Morally Due Tue May 3 at 3:30PM. Dead Cat May 5 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) Let $VDW(k,c)$ be the statement {\it There exists $W=W(k,c)$ such that for all $\COL\colon [W]\into[c]$ there exists $a,d\ge 1$ such that $$a, a+d, \ldots, a+(k-1)d \hbox{ are the same color }.$$ } Let $W(k,c)$ be as in the statement. AND NOW FOR THE PROBLEM Assume $(\forall c)[VDW(9,c)]$. Prove $\VDW(10,2)$. Your proof should give an upper bound on $W(10,2)$ as a function of $W(9,c)$. \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (50 points) In this problem you will do PART of the proof of $(\forall k)[\PVDW(x,x^2,x^2+x,\ldots,x^2+kx)]$. \begin{enumerate} \item (20 points) State carefully the LEMMA that will imply $(\forall k)[\PVDW(x^2,x^2+x,\ldots,x^2+kx)]$. \item (30 points) Prove carefully the BASE CASE of that lemma. \item (0 points, but good for your enlightenment) Prove the Induction Step of the lemma. \end{enumerate} \vfill \centerline{\bf GOTO NEXT PAGE} \newpage \end{enumerate} \end{document}