\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 03} \centerline{Morally Due Tue Feb 15 at 3:30PM. Dead Cat Feb 17 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) Give a well written complete proof of Mileti's proof of the 2-ary Can Ramsey Theorem. (I did the first two steps in class, but you will need to include those as well.) \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (35 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 countable infinite set of points in the plane, no three colinear. 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 $\COL\colon\binom{N}{3}\into \R$ via $\COL(i,j,k)$ is the area of the triangle created by $p_i,p_j,p_k$. The number of reals used is countable so we can apply Can Ramsey. Hence there exists $H\subseteq \N$, $|H|=\infty$, $H$ is $A$-homog for some $A\subseteq \{1,2,3\}$. 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 (30 points) {\bf Point of this Problem} The first day of class we proved that no matter how you color the edges of $K_6$ there will be a monochromatic triangle. What about $K_5$? It turns out that there IS a coloring of $K_5$ with NO mono triangles. But how common is that? In this problem you will generate 1000 random 2-colorings of the edges of $K_5$ and COUNT how many have 0 mono triangle, 1 mono triangle, $\ldots$, 10 mono triangles. You will generate these colorings 9 different ways. Each time you do it you will count how many of the colorings had 0 mono triangles, 1 mono triangle, $\ldots$, 9 mono triangles. For $0\le i\le 10$, $n_i$ will be the number that have $i$ mono triangles. NOTE: All we want to hand in will be the table of data, and some speculation about theorems, NOT the code itself. On the next page IS the problem formally. \vfill \centerline{\bf GO TO NEXT PAGE} \newpage {\bf ONE} (0 points but you need to do this for later parts) Write a program that will take input $0\le p\le 1$ and randomly assign colors to the edges of $K_5$ with each edge being RED with prob $p$ and BLUE with prob $1-p$. (You might want to use 0 and 1 instead of RED and BLUE since computers operate that way.) \smallskip {\bf TWO} (0 points but you need to do it for later parts) Write a program that will, given a 2-coloring of $K_5$, count how many monochromatic triangles it has. {\bf THREE} (0 points but you need this for later parts) Write a program that does the following (I use psuedocode.) \begin{enumerate} \item[ ] For $p= 0.1, 0.2, \ldots, 0.9$ \begin{enumerate} \item $n_0=0$, $n_1=0$, $\ldots$, $n_{10}=0$. (Recall that $n_i$ will be the number of colorings that have $i$ mono triangles. Initially this is 0.) \item For $i=1$ to 1000 \begin{enumerate} \item Randomly color the edges of $K_5$ by coloring RED with prob $p$ and BLUE with prob $1-p$. (You may want to use colors 0 and 1 instead.) \item Find $j$, the number of mono triangles. \item $n_j = n_j+1$. \end{enumerate} \end{enumerate} \end{enumerate} {\bf FOUR} (30 points) Use your program to produce the a table of data The table should look like what is below except that (1) I made up the numbers, and (2) your table should not have any DOT DOT DOT in it, it should have all the numbers. $\begin{array}{|c|ccccccccccc|} \hline p & n_0 & n_1 & n_2 & n_3 & n_4 & n_5 & n_6 & n_7 & n_8 & n_9 & n_{10} \cr \hline 0.1 & 0 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 5 & 5 \cr 0.2 & 0 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 10 & 7 & 3 \cr \vdots & \vdots & \vdots & \vdots & \vdots & \vdots& \vdots & \vdots &\vdots & \vdots & \vdots & \vdots \cr 0.9 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \cr \hline \end{array}$ \smallskip {\bf FIVE} (0 points) Looking at the data formulate a conjecture about colorings of $K_5$. Prove your conjecture. \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}