\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 13} \centerline{\bf Morally Due Tue May 10 at 3:30PM. Dead Cat May 12 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) In this problem we look at the problem of dividing 8 muffins for 7 people so that everyone gets $\frac{8}{7}$. Recall that $f(8,7)$ is the size of the smallest piece in an optimal protocol. \begin{enumerate} \item (10 points) Use the Floor-Ceiling Formula to get an upper bound on $f(8,7)$. Express as both a fraction and in decimal up to 3 places. \item (25 points) Use the HALF method to show that $f(8,7)\le \frac{5}{14}$. You can assume that each muffins is cut into 2 pieces so that there are 16 pieces. You can assume that nobody gets just 1 share (if they did then they would have 1 muffins, but they should get $\frac{8}{7}>1$). \item (15 points) Give a PROTOCOL that achieves the bound $\frac{5}{14}$. We give the format we want for the $f(5,3)$ problem. Do a similar format. $f(5,3)\ge\frac{5}{12}$: \begin{enumerate} \item Divide 1 muffins $(\frac{6}{12},\frac{6}{12})$. \item Divide 4 muffins $(\frac{5}{12},\frac{7}{12})$. \item Give 2 students $\{\frac{6}{12}, \frac{7}{12}$, $\frac{7}{12}\}$. \item Give 1 students $\{\frac{5}{12}, \frac{5}{12}$, $\frac{5}{12}\}$. \end{enumerate} \end{enumerate} \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (50 points) Reread the proof that, for all $c$, there exists a graph $G_c$ such that $\chi(G_c)=c$ and $g(G_c)=6$. Let $M_c$ be the number of notes in $G_c$. Write a recurrence for the $M_c$ in terms of $M_{c-1}$. \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (0 points-- For your own Enlightenment) Reread the proof that, for all $c$, there exists a graph $G_c$ such that $\chi(G_c)=c$ and $g(G_c)=9$. It used PVDW and {\it easy number theory}. Write down a theorem in Number Theory whose truth would show that the construction yields $g(G_c)\ge 12$. \vfill \centerline{\bf GOTO NEXT PAGE} \newpage \item (10 points) For each of the following say if its an APPLICATION!, an application'' or a are you kidding me?'' Explain why you think so. \begin{enumerate} \item Using Ramsey Theory to prove the Bolzano-Weierstrass Theorem. \item Using Hypergraph Ramsey to prove Can Ramsey. \item Using Ramsey to assist program checkers to prove programs halt. \item Using Can Ramsey to show that for all infinite sets of points in the plane there is an infinite subset where all distances are distinct. \item Using Ramsey to show that if $(X,\preceq)$ is a wqo then every sequence has an infinite increasing subsequence. \item Using Ramsey to show that, for all $k$, there exists $N$, so given any $N$ points in the plane, no 3 colinear, there is a set of $k$ that form a convex $k$-gon. \item Using Ramsey to show that in the language of graphs (or colored $\le a$-ary hypergraphs) the spectrum problem for $\exists^* \forall^*$ sentences is decidable. \item Using Ramsey to show that if the universe is big enough then tables should be sorted. \item Using Poly VDW to construct graphs $G$ with $\chi(G)=c$ and $g(G)=9$. \item Using VDW to show there that, for all $p$, there exists long sequences of consecutive squares mod $p$ (if I get to it). \end{enumerate} \end{enumerate} \end{document}