\documentclass[12pt,ifthen]{article} \usepackage{amsmath} \usepackage{amssymb} %\usepackage{html} \usepackage{hyperref} \newcommand{\degg}{{\rm deg}} \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 05} \centerline{Morally Due Tue March 1 at 3:30PM. Dead Cat March 3 at 3:30PM} \begin{enumerate} \item (0 points) What is your name? Write it clearly. When is the take-home midterm due? \item (30 points) Prove the following (the finite 3-hypergraph Ramsey Theorem) by using the infinite 3-hypergraph Ramsey Theorem. {\it For all $k,c$ there exists $n$ such that for all $\COL\colon\binom{[n]}{3}\into[c]$ there exists a homog set of size $k$. } \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (30 points) We will look at the following statement which we call FCR (Finite Can Ramsey) {\it For all $k,c$ there exists $n$ such that for all $\COL\colon\binom{[n]}{2}\into\omega$ either there is a homog set of size $k$ OR a min-homog set of size $k$ OR a max-homog set of size $k$ OR a rainbow set of size $k$.} \begin{enumerate} \item (10 points) TRY to prove FCR from the infinite Ramsey Theorem. YOU WILL FAIL. Where does it fail? \item (20 points) Prove FCR somehow (Hint: DO NOT try to finitize Mileti's proof. That can be done but is messy.) \end{enumerate} \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (40 points) In this problem we will look at the following problems: {\it Given $X\subseteq \R^1$ (or $\S^1$ or $\R^2$) of size $n$ show there is a large subset of size $\Omega(f(n))$ (you will figure out the $f$) where all of the distances are different. ($S^1$ is any circle.) } We will NOT use Can Ramsey. Let $X$ be a set of points (could be in $\R^1$ or $S^1$ or $\R^2$). Let $M\subseteq X$. $M$ is {\it d-maximal} if (1) every pair of distances is different and (2) for all $p\in X-M$, statement (1) is false for $M\cup \{p\}$. (Note that we DO NOT have the color-degree bound we had in the past.) \begin{enumerate} \item (10 points) Find an increasing function $f$ such that the following is true: {\it If $X\subseteq \R^1$ is a set of $n$ points then every d-maximal set is of size $\Omega(f(n))$.} \item (15 points) Find an increasing function $g$ such that the following is true: {\it If $X\subseteq \S^1$ is a set of $n$ points then every d-maximal set is of size $\Omega(g(n))$.} \item (15 points) Find an increasing function $h$ such that the following is true: {\it If $X\subseteq \R^2$ is a set of $n$ points then every d-maximal set is of size $\Omega(h(n))$.} \end{enumerate} \end{enumerate} \end{document}