\documentclass[12pt,ifthen]{article} \usepackage{amsmath,amssymb} %\usepackage{html} %\usepackage{url} \usepackage{hyperref} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\into}{{\rightarrow}} \newcommand{\st}{\mathrel{:}} \newcommand{\finv}{f^{-1}} \newcommand{\COL}{\mathrm{COL}} \begin{document} \centerline{\bf Homework 6, Morally Due Tue April 14, 2020, 3:30PM} COURSE WEBSITE: \url{http://www.cs.umd.edu/~gasarch/858/S18.html} \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. \item (50 points) In the March 31 recording I gave three proofs of the following theorem: {\bf Theorem:} For all $k$ there exists $n$ such that, for any $n$ points in the plane no three colinear, there exists $k$ points that form a convex $k$-gon. One proof used 5-ary Ramsey. One proof used 3-ary Ramsey and the coloring $COL(x,y,z)$ is RED if the number of points inside the x-y-z triangle is RED is even, BLUE if its odd. One proof used 3-ary Ramsey and the following coloring: let $p_1,\ldots,p_n$ be the points (the ordering does not correspond to anything geometric, but we need SOME way to order the points). $COL(p_i,p_j,p_k)$ with \$i