\documentclass[11pt]{article} \usepackage{fullpage} \usepackage{url} \usepackage{amsmath} % This is a file of macros and definitions that may come up in % ANY LATEX paper. Typically I'll use this file and some other file in the % directory of the paper, this one for general math things, that one for % things specific to that paper. % % font's used and general paper things. \font\tenrm=cmr10 \font\ninerm=cmr9 \font\eightrm=cmr8 \font\sevenrm=cmr7 % \font\title=cmbx10 scaled \magstep1 % extra big title font \font\ss=cmss10 % used by \proof \font\smallcaps=cmcsc10 % used to label Theorems, etc. % imhibit black bars on overflows % \overfullrule=0pt % % today's date % % % English words that I always italizice in papers. % Some words that appear in math mode alot that I wasn roman % \newcommand{\AREA}{{\rm AREA}} \newcommand{\ALG}{{\rm ALG}} \newcommand{\ACK}{{\rm ACK}} \newcommand{\VC}{{\rm VC}} \newcommand{\AC}{{\rm AC}} \newcommand{\ZL}{{\rm ZL}} \newcommand{\WOP}{{\rm WOP}} \newcommand{\tree}{{\rm tree}} \newcommand{\SUBSEQ}{{\rm SUBSEQ}} \newcommand{\minor}{{\preceq_m}} \newcommand{\minorp}{{\preceq_m'}} \newcommand{\cminor}{{\preceq_{\rm c\hbox{-}m}}} \newcommand{\GM}{{\rm GM}} \newcommand{\TREE}{{\rm TREE}} \newcommand{\R}{\rm R} \newcommand{\B}{\rm B} \newcommand{\G}{\rm G} \newcommand{\Y}{\rm P} \newcommand{\IO}{\exists^\infty} \newcommand{\ceil}[1]{\left\lceil {#1}\right\rceil} \newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor} \newcommand{\into}{{\rightarrow}} \newcommand{\D}{{\sf D}} \newcommand{\N}{{\sf N}} \newcommand{\Z}{{\sf Z}} \newcommand{\Q}{{\sf Q}} \newcommand{\reals}{{\sf R}} \newcommand{\qd}{{\rm qd}} \newcommand{\GE}[1]{{\equiv^{G}_{#1}}} \newcommand{\TE}[1]{{\equiv^{T}_{#1}}} \newcommand{\TES}[1]{{\equiv^{T}_{#1,{\rm S}}}} \newcommand{\TEB}[1]{{\equiv^{T}_{#1,{\rm BOOL(S)}}}} \newcommand{\LL}{{\cal{L}}} \newcommand{\onepe}{1+\epsilon} \newcommand{\oneme}{1-\epsilon} \newcommand{\cool}{cool} \newcommand{\nze}{numzeros} \newcommand{\non}{numones} \newcommand{\LR}[2]{ {\rm LR}_{{#2}}({#1})} \newcommand{\LH}[2]{ {\rm LH}({#1};{#2})} \newcommand{\bbb}{{\gamma}} \newcommand{\COL}{{\rm COL}} \newcommand{\card}[1]{\#(#1)} \newcommand{\st}{\mathrel{:}} \newcommand{\fhat}{{\hat{f}}} \newcommand{\Ahat}{{\hat{A}}} \begin{document} \centerline{\bf Homework 11, Morally Due 12:30PM, Tue Apr 28 2026} \begin{enumerate} \item (0 points) What is your name? \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (35 points) \begin{enumerate} \item (10 points) Show that $\binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}$ with a combinatorial proof. That is, show that the Right Hand Side solves the problem {\it how many ways can I pick $k$ Ramsey Theorists from a set of $n$ Ramsey Theorists}. (You will use this in the next part.) \item (15 points) Show that, for all $a,b\ge 2$, $R(a,b) \le \binom{a+b-2}{a-1}$. \item (10 points) Use the result $R(a,b)\le \binom{a+b-2}{a-1}$ to get an asymptotic upper bound on $R(k)$ that is LESS THAN $2^{2k-1}$. \end{enumerate} \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (30 points) {\bf Notation} $\N^{\ge 2}$ means the set $\{2,3,4,\ldots\}$. {\bf Def} Let $G=(V,E)$ be a graph. $D\subseteq V$ is a {\it dominating set (DS)} if $$(\forall u\in V)[u\in D \vee (\exists v\in D)[(u,v)\in E].$$ Every graph has a DS of size $n$: $D=V$. We do better! You will prove: {\it There exists a function $\alpha\colon \N^{\ge 2} \into (0,1)$ such that, a) For every $d\in\in\N^{\ge 2}$, $\alpha(d+1)<\alpha(d)$ (so $\alpha(d)$ is \textbf{STRICTLY DECREASING}). b) For every graph with min degree $\ge d$ there is a dominating set of size $\le \alpha(d)n$. } On the next page we will state the theorem with the function $\alpha$ and sketch the proof. YOU will fill in the details and find a function $\alpha$ that works. We guide this with a series of embedded questions. \newpage \noindent {\bf Thm} There exists a function $\alpha\colon \N^{\ge 2}\into (0,1)$ such that the following hold: \begin{itemize} \item $\alpha$ is strictly DECREASING. \item If $G=(V,E)$ is a graph on $n$ vertices with min degree $\ge d$ then $G$ has a dominating set of size $\le \alpha(d)n$. \end{itemize} \noindent {\bf Proof Sketch} Let $p$ be a probability to be determined by YOU later. Pick $X\subseteq V$ as follows: For every $v\in V$ choose $v$ with probability $p$. \smallskip \begin{enumerate} \item (0 points but you need it for later) What is $E(|X|)$? It will be a function of $n,p$. \item (0 points but you need it for later) Let $Y\subseteq V-X$ be the vertices that DO NOT have an edge to an element of $X$. Formally $$Y= \{ y\in V-X \colon (\forall x\in X)[(x,y)\notin E].$$ Give an upper bound on $E(|Y|)$. It will be a function of $n,d,p$. Note that $X\cup Y$ is a dominating set. We later pick $p$ so that $|X\cup Y|$ is small. \item (0 points but you need it for later) What is $E(|X\cup Y|)$? (Hint: This is very easy by the linearity of expectation.) \item (30 points) Pick $p$ to make $E(|X\cup Y|)$ smaller than $n$ then give the function $\alpha$. (Hint: Find an upper bound on $E(|X\cup Y|)$ and minimize that bound. Use that $(1-p)\le e^{-p}$.) \noindent {\bf End of Proof of Sketch} \end{enumerate} \vfill \centerline{\bf GO TO NEXT PAGE} \newpage \item (35 points) \begin{enumerate} \item (0 points but you need to do it for the rest of the problem) Write a program that will, given $n$, generate a random 2-coloring of $\binom{[n]}{2}$ by, for each edge, color it RED with prob $\frac{1}{2}$ and BLUE with prob $\frac{1}{2}$. \item (0 points but you need to do it for the rest of the problem) Write a program that will, given $n$ and a 2-coloring of $\binom{[n]}{2}$, count how many mono $K_3$'s there are. \item (0 points but you need to do it for the rest of the problem) This problem just puts the two programs together. Write a program that will, given $n$, generate a random 2-coloring of $\binom{[n]}{2}$ by, for each edge, color it RED with prob $\frac{1}{2}$ and BLUE with prob $\frac{1}{2}$ and then OUTPUTS the number of mono $K_3$'s. \item (0 points but you need to do it for the rest of the problem) This problem mostly uses the last program. Write a program that will, given $n$, run the last program 100 times and gather up the number-of-triangles. Then just output the MIN and the MAX. \item (35 points) Write a program that outputs a table with \begin{itemize} \item $n$ going from 4 to 30 \item Min number of mono $K_3$'s from last program. \item Max number of mono $K_3$'s from last program. \item How much bigger is the Max from what we are guaranteed by the theorems in class. We call this {\it over}. \end{itemize} Here is the format of the table (I made up the numbers and only give the first 3 rows): \begin{tabular}{|c|c|c|c|} \hline $n$ & min & max & over \cr \hline 4 & 1 & 2 & 2 \cr 5 & 1 & 2 & 1 \cr 6 & 2 & 8 & 3 \cr \hline \end{tabular} \item (0 points) Make a conjecture about what happens for a random coloring. \end{enumerate} \end{enumerate} \end{document}