\documentclass[12pt,ifthen]{article} \usepackage{amsmath} \usepackage{amssymb} %\usepackage{html} \usepackage{hyperref} \newcommand{\degg}{{\rm deg}} \newcommand{\COL}{{\rm COL}} \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 04} \centerline{Morally Due Tue Feb 22 at 3:30PM. Dead Cat Feb 24 at 3:30} \begin{enumerate} \item (0 points) What is your name? Write it clearly. When is the take-home midterm due? \item (40 points) Assume $(X,\preceq_X)$ and $(Y,\preceq_Y)$ are wqo. Consider the ordering $(X\times Y,\preceq)$ where $\preceq$ is defined as $$(x_1,y_1) \preceq (x_2,y_2) \hbox{ iff } x_1 \preceq_X x_2 \hbox{ AND } y_1\preceq_Y y_2.$$ Show that $(X\times Y,\preceq)$ is a wqo. \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (50 points) Assume $(X,\preceq)$ is a wqo. Let $\PF(X)$ be the set of finite subsets of $X$. Let $\preceq'$ be the following order on $\PF(X)$. Let $Y,Z\in \PF(X)$. $Y\preceq' Z$ iff there exists a FUNCTION $f:Y\into Z$ such that $(\forall y\in Y)[y\preceq f(y)]$. \begin{enumerate} \item (20 points) Prove or disprove: $(\PF(X),\preceq')$ is a wqo. \item (15 points) Modify $\preceq'$ such that the function $f$ has to be injective (also called 1-1). Prove or disprove: $(\PF(X),\preceq')$ is a wqo. \item (15 points) Modify $\preceq'$ such that the function $f$ has to be surjective (also called onto). Prove or disprove: $(\PF(X),\preceq')$ is a wqo. \end{enumerate} \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (10 points) GOTO my webpage of funny music and GOTO the section on Math Songs \url{https://www.cs.umd.edu/~gasarch/FUN//funnysongs.html} \begin{itemize} \item Listen to the Bolzano Weirstrauss rap- or as much of it as you can stand. Comment on it. \item Pick ANY OTHER math song AT RANDOM and listen to it. Is it better than the BW rap (hint: YES). Comment on it. \end{itemize} \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (Extra Credit- NOT towards your grade but towards a letter I may one day write for you) (This will look like a prior extra credit but it's a new problem.) {\it Definition} A {\it bipartite} graph is a graph with vertices $A\cup B$ and the only edges are between vertices of $A$ and vertices of $B$. $A$ and $B$ can be the same set. We denote a bipartite graph with a 3-tuple $(A,B,E)$. {\it Notation} $K_{n,m}$ is the bipartite graph $([n],[m],[n]\times [m])$. {\it Notation} $K_{\N,\N}$ is the bipartite graph $(\N,\N,\N \times\N)$. {\it Definition} If $COL$ is a $c$-coloring of the edges of $K_{\N,\N}$ then $(H_1,H_2)$ is a homog set if $c$ restricted to $H_1\times H_2$ takes on only 1 value (I changed the wording on this so I can generalize it later.) {\bf RECALL} In a prior extra credit problem we DISPROVED the following: {\it For every 2-coloring of the edges of $K_{\N,\N}$ there exists $H_1$, $H_2$ infinite such that $(H_1,H_2)$ is a homog set. } In other words we showed the following: {\it There IS a 2-coloring of the edges of $K_{\N,\N}$ such that there is NO $H_1$, $H_2$ infinite such that $(H_1,H_2)$ is a homog set. } This inspires the following definition. {\it Definition} Let $d\le c$. If $COL$ is a $c$-coloring of the edges of $K_{\N,\N}$ then $(H_1,H_2)$ is a $d$-homog set if $c$ restricted to $H_1\times H_2$ COL takes on $\le d$ values. SO to recap- we could have a 2-coloring of the edges of $K_{\N,\N}$ where there is no 1-homog set. But there is clearly a 2-homog set, namely $(\N,\N)$. {\bf And now FINALLY the problem:} {\it For ever $k\ge 3$ Prove or disprove: For every $k$-coloring of the edges of $K_{\N,\N}$ there exists $H_1$, $H_2$ infinite such that $(H_1,H_2)$ is a 2-homog set. } \end{enumerate} \end{document}