\documentclass[12pt,ifthen]{article} \usepackage{amsmath} \usepackage{amssymb} %\usepackage{html} \usepackage{hyperref} \newcommand{\degg}{{\rm deg}} \newcommand{\JULY}{{\rm JULY}} \newcommand{\PH}{{\rm PH}} \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 07} \centerline{Morally Due Tue March 29 at 3:30PM. Dead Cat March 31 at 3:30} {\bf IN THIS HW WHENEVER I SAY A SET OF POINTS IN THE PLANE'' I MEAN THAT THEY HAVE NO THREE COLINEAR.} \begin{enumerate} \item (0 points) What is your name? Write it clearly. When is the take-home final due? \item (35 points) Let $N(k)$ be the least $n$ such that for all sets of $n$ points there is a subset of $k$ of them that form a convex $k$-gon. We begin a proof that $N(k)$ exists and you need to finish it. {\it We show that $n=R_3(k)$ suffice. Let $X$ be a set of $n=R_3(k)$ points in the plane. Let the points be $p_1,p_2,\ldots,p_n$. Color $(p_i,p_j,p_j)$ (with $iMIN(H)$, \item $\COL$ restricted to $\binom{H}{2}$ is constant. \end{itemize} Find a number $A$ such that you can prove $\JULY(1)\le A$. (I have a proof with $A=8$ but given that the original version of this problem was incorrect, I am phrasing it this way so it can't go wrong.) \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (30 points) Recall: {\it If $n\equiv 1 \pmod 2$ then for any $\COL\colon\binom{[n]}{2}\into[2]$ there exists at least $$\frac{n^3}{24} -\frac{n^2}{4} + \frac{5n}{24}$$ monochromatic $K_3$'s.} We will vary this in two ways. \begin{enumerate} \item (15 points) Find a function $f$ such that the followings is true: {\it If $n\equiv 0 \pmod 2$ then for any $\COL\colon\binom{[n]}{2}\into[2]$ there exists at least $f(n)$ monochromatic $K_3$'s.} Prove your result. \item (15 points) We are interested in what happens if you have THREE colors. Do some empirical studies to try to find a function $f$ such that the following holds: {\it If $\COL\colon\binom{[n]}{2}\into[3]$ then there exists at least $f(n)$ monochromatic $K_3$'s. ($f(n)$ an be approximate. For example, if the problem was for 2-coloring then $f(n)$ could be $\frac{n^3}{24}$.) } (HINT: Use the code you wrote for the midterm; however, only use the case of $p_1=p_2=p_3=\frac{1}{3}$.) \item (Extra Credit, 0 points) PROVE a result along the lines of: {\it If $n$ satisfies condition YOU FILL IN and $\COL\colon\binom{[n]}{2}\into[3]$ then there exists at least $f(n)$ monochromatic $K_3$'s.) (HINT: Use the code you wrote for the midterm; however, only use the case of $p_1=p_2=p_3=\frac{1}{3}$.) } \end{enumerate} \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (Extra Credit, but THINK ABOUT IT. WARNING- I have not done this problem) {\it Let $X$ be an infinite set of points $p_1,p_2,p_3,\ldots$. Let $\COL\binom{\N}{3}\into \omega$ be defined as follows: $$\COL(i,j,k) = \hbox{ the number of points inside the (i,j,k) triangle.}$$ Apply the 3-ary Can Ramsey Theorem to this Coloring. NOW WHAT? } \item (Extra Credit, but THINK ABOUT IT--WARNING: the way I know how to do this is based on material you have not seen) We want to write a sentence $\phi$ in the language of graphs such that $G\models \phi$ IFF $G$ has an even number of vertices. Is this possible? \end{enumerate} \end{document}