\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 06} \centerline{Morally Due Tue March 8 at 3:30PM. Dead Cat March 10 at 3:30} \begin{enumerate} \item (0 points) What is your name? Write it clearly. When is the take-home midterm due? \item (35 points) Let $R_a(k)$ be the least $n$ such that \centerline{\it for all $\COL\colon\binom{[n]}{a}\into[2]$ there exists a homog set of size $k$.} For this problem assume $R_2(k)\le 2^{2k}$ (which is true). In class I sketched the beginning of the proof that $R_3(k) \le 2^{2^{O(k)}}$. For this problem give a complete rigorous proof. \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (35 points) Prove the following: {\it For all $k$ there exists $n$ such that for all $COL:\binom{\{k,\ldots,n\}}{1}\into \omega$ there exists either \begin{itemize} \item a LARGE homog set, or \item a LARGE rainbow set (all the numbers are colored differently). \end{itemize} } \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (30 points) Prove the following: {\it For all $k$ there exists $n$ such that for all $COL:\binom{\{k,\ldots,n\}}{2}\into [100]$ there exists an $H\subseteq [n]$ such that \begin{itemize} \item $H$ is a homog set, and \item $|H|\ge 2^{2^{\min(H)}}$. \end{itemize} } \end{enumerate} \end{document}