\documentclass[12pt,ifthen]{article} \usepackage{amsmath} \usepackage{amssymb} %\usepackage{html} \usepackage{hyperref} \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 11} \centerline{\bf Morally Due Tue April 26 at 3:30PM. Dead Cat April 28 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) Find a value of $m<50$ such that the following holds, and prove it. Prove it from first principles--- that is, your proof should not refer to the slides or any other source. {\it For all 3-colorings of the $4\times m$ grid, there exists a mono rectangle.} \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (50 points) For this problem we can assume the following is known: {\it For all $c\ge 1$ there exists $L=L(c)$ such that for all $c$-colorings of the $L(c)\times L(c)$ grid there exists a monochromatic isocles $L$.} Let a big-base-$L$ be the same shape as the following four points: $(0,0)$, $(d,0)$, $(2d,0)$, and $(0,d)$. And NOW for our problem: {\it Show that there exists $LL$ such that, for all 2-colorings of the $LL\times LL$ grid there exists a monochromatic big-base-$L$.} Feel free to use PICTURES in your proof. \end{enumerate} \end{document}