\documentclass[12pt,ifthen]{article} \usepackage{amsmath} \usepackage{amssymb} %\usepackage{html} \usepackage{hyperref} \newcommand{\N}{{\sf N}} \newcommand{\Z}{{\sf Z}} \newcommand{\R}{{\sf R}} \newcommand{\into}{{\rightarrow}} \newcommand{\ceil}[1]{\lceil #1 \rceil} \usepackage{amsthm} \newtheorem{theorem}{Theorem} \begin{document} \centerline{\bf Homework 1} \centerline{Morally Due Tue Feb 1 at 3:30PM} COURSE WEBSITE: \url{http://www.cs.umd.edu/~gasarch/COURSES/752/S22/index.html} (The symbol before gasarch is a tilde.) \begin{enumerate} \item (0 points) What is your name? Write it clearly. When is the take-home midterm due? {\bf Learn LaTeX if you don't already know it} \item (20 points) \begin{enumerate} \item (9 points) Prove that for every $c$, for every $c$ coloring of $\binom{\N}{2}$, there is a homogenous set USING a proof similar to what I did in class. \item (9 points) Prove that for every $c$, for every $c$ coloring of $\binom{\N}{2}$, there is an infinite homogenous set USING induction on $c$. \item (2 points) Which proof do you like better? Which one do you think gives better bound when you finitize it? \end{enumerate} \vfill \centerline{\bf GOTO NEXT PAGE} \newpage \item (20 points) Prove the following theorem rigorously (this is the infinite $c$-color $a$-ary Ramsey Theorem): \noindent {\bf Theorem} For all $a\ge 1$, for all $c\ge 1$, and for all $c$-colorings of $\binom{\mathbb{N}}{a}$, there exists an infinite set $A\subseteq \mathbb{N}$ such that $\binom{A}{a}$ is monochromatic ($A$ is an infinite homogeneous set). \noindent {\bf End of Statement of Theorem} The proof should be by induction on $a$ with the base cases being $a=1$. You need to prove the base case. \vfill \centerline{\bf GOTO NEXT PAGE} \newpage \item (20 points) Lets apply Ramsey Theory! \begin{enumerate} \item (20 points) Let $$x_1,x_2,x_3,\ldots,$$ be an infinite sequence of distinct reals. Consider the following coloring of $\binom{N}{2}$. Let $ix_j$}\cr \end{cases} If you apply Ramsey Theory to this coloring you get a theorem. State that theorem cleanly. \item (0 points, but REALLY try to do it) Prove the theorem you stated in Part a WITHOUT USING Ramsey Theory. \item (0 points, but REALLY do it) Which proof do you prefer, the one that use Ramsey Theory or the one that didn't? \end{enumerate} \vfill \centerline{\bf GOTO NEXT PAGE} \newpage \item (20 points) Lets apply Ramsey Theory! \begin{enumerate} \item (9 points) Let $$x_1,x_2,x_3,\ldots,$$ be an infinite sequence of points in $\R^2$. (NOTE- these are points in $\R^2$, not reals. So this is a different setting from the prior problem.) Consider the following coloring of $\binom{N}{2}$. $$COL(i,j) = \begin{cases} RED & \hbox{if d(x_i,x_j)>1}\cr BLUE& \hbox{if d(x_i,x_j)<1}\cr \end{cases}$$ If you apply Ramsey Theory to this coloring you get a theorem. State that theorem cleanly. \item (9 points) Prove the theorem you stated in Part a WITHOUT USING Ramsey Theory. \item (2 points) Which proof do you prefer? \end{enumerate} \vfill \centerline{\bf GOTO NEXT PAGE} \newpage \item (Extra Credit- NOT towards your grade but towards a letter I may one day write for you) {\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 $B$. $A$ an $B$ can be the same set. We denote a biparatite 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 $H_1\times H_2$ is constant. And now FINALLY the problem. Prove or disprove: {\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. } \newpage \item (Extra Credit- NOT towards your grade but towards a letter I may one day write for you) Recall that the infinite Ramsey Theorem for 2-coloring the edges of a graph: {\it For all colorings $COL:\binom{\N}{2}\into [2]$ there exists an infinite homogenous set $H\subseteq \N$.} What if we color $\Z$ instead of $\N$? If all we want is an {\it infinite homogenous set} then the exact same proof works---or you could just restrict the coloring to $\binom{\N}{2}$. But what if we want an infinite $H\subseteq \Z$ that has {\it the same order type as $\Z$}? {\bf Definition} If $(L_1,<_1)$ and $(L_2,<_2)$ are ordered sets then they are {\it order-equivalent} if there is a bijection $f$ from $L_1$ to $L_2$ that preserves order. That is, $x<_1 y$ iff $f(x) <_2 f(y)$. And now FINALLY the problem: Prove or disprove: {\it For all colorings $COL:\binom{\Z}{2}\into [2]$ there exists a set $H\subseteq \Z$ that is order-equiv to $\Z$ and is homogenous.} \end{enumerate} \end{document}