\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 10} \centerline{\bf Morally Due Tue April 19 at 3:30PM. Dead Cat April 21 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) \begin{enumerate} \item (0 points but some of it will help in later parts) READ my NOTES on Duplicator Spoiler games. They are on the slides website next to my slides. \item (25 points) Define a Duplicator Spoiler game where the two structures are UNDIRECTED GRAPHS rather than LINEAR ORDERINGS. \item (25 points) Show that, for all $n\in\N$, for all $k\in\N$, with \$k