\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{\ceil}[1]{\lceil #1 \rceil} \usepackage{amsthm} \newtheorem{theorem}{Theorem} \begin{document} \centerline{\bf Homework 2} \centerline{Morally Due Tue Feb 15 at 3:30PM. Dead Cat Feb 17 at 3:30} \newif{\ifshowsoln} \showsolntrue % comment out to NOT show solution inside \ifshowsoln and \fi blocks. \begin{enumerate} \item (0 points) What is your name? Write it clearly. When is the take-home midterm due? \item (35 points) Look at the slides on the Can Ramsey Theorem. Look at the proof that uses 4-ary Ramsey. RECAP OF THAT PROOF FOR OUR PURPOSES: Given a coloring $\COL:\binom{N}{2}\into [\omega]$ we created $\COL':\binom{N}{4}\into [16]$. We then applied 4-ary Ramsey Theory to get a homog set $H$ (relative to $\COL'$). We show that whatever color the homog set was we found an infinite subset of $H$ that was with respect to $\COL$ either (a) homog, (b) max-homog, (c) min-homog, or (d) rainbow. OKAY, now for our problem. Look at the case where there is an infinite homog (using coloring $\COL'$) $H$ such that $$(\forall x_1 < x_2 < x_3 < x_4 \in H)[ \COL(x_2,x_3)=\COL(x_1,x_4)].$$ Show that $H$ (or perhaps some infinite subset of it) is homog with respect to $\COL$. (I am asking you to do one of the cases I skipped.) \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (35 points) Before the proof of Can Ramsey that used 3-ary Ramsey we proved the following: {\it Assume $X$ is infinite and $\COL:\binom{X}{2}\into [\omega]$. Assume that, for all $x\in X$ and colors $c$, $\degg_c(x)\le 1$. Let $M$ be a MAXIMAL rainbow set. Then $M$ is infinite.} In this problem you will come up with and prove a FINITE version of this theorem. Fill in the function $f(n)$ and prove the following: {\it Assume $|X|=n$ and $\COL:\binom{X}{2}\into [\omega]$. Assume that, for all $x\in X$ and colors $c$, $\degg_c(x)\le 1$. Let $M$ be a MAXIMAL rainbow set. Then $|M|\ge \Omega(f(n))$.} \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (30 points) In this problem we have a part of a proof, but want a theorem. Fill in the BLANK and the BLAH BLAH to get a theorem OF INTEREST. \noindent {\bf Theorem} Let $X$ be a countably infinite set of points in the plane. Then there exists $Y\subseteq X$, $|Y|=\infty$, such that BLANK. \noindent {\bf Proof} Order the points in $X$ arbitrarily, so $$X = \{ p_1, p_2, p_3, \ldots \}.$$ Define a coloring $\COL\colon\binom{N}{2}\into \R$ via $\COL(i,j)=|p_i-p_j|$. The number of reals used is countable so we can apply Can Ramsey. Hence there exists $H\subseteq \N$, $|H|=\infty$, $H$ is either homog, min-homog, max-homog, or rainbow. Look at the set of points $$Y=\{p_i \colon i\in H\}.$$ Then BLAH BLAH so $Y$ is BLANK. \noindent {\bf End of Proof of Theorem} \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (Extra Credit) Give a well written clean proof of 3-ary Can Ramsey. There are three ways to do this. The more ways you do, the more extra credit you get! \begin{enumerate} \item Use some $a$-ary Ramsey Theorem and lots of cases (with good notation you can consolidate them), and all cases easy. \item Use some $a$-ary Ramsey Theorem with fewer cases than the proof suggested in Part 1 (with good notation you can consolidate them), and the rainbow case will need a version of maximal sets. \item Use a Mileti-style proof. Note that 2-ary Mileti used 1-ary Can Ramsey. Similarly, 3-ary Mileti will use 2-ary Can Ramsey. It will be similar to the proof of 3-ary Ramsey from 2-ary Ramsey. \end{enumerate} \end{enumerate} \end{document}