\documentclass[12pt,ifthen]{article}
\usepackage{amsmath}
\usepackage{amssymb}
%\usepackage{html}
\usepackage{hyperref}
\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 1} 

\centerline{Morally Due Tue Feb 1 at 3:30PM}

COURSE WEBSITE:

\url{http://www.cs.umd.edu/~gasarch/COURSES/752/S22/index.html}

(The symbol before gasarch is a tilde.)



\begin{enumerate}

\item
(0 points)
What is your name?  Write it clearly.
When is the take-home midterm due?
{\bf Learn LaTeX if you don't already know it}

\item
(20 points)
\begin{enumerate}
\item
(9 points)
Prove that for every $c$, for every $c$ coloring of
$\binom{\N}{2}$, there is a homogenous set USING a proof
similar to what I did in class.
\item
(9 points)
Prove that for every $c$, for every $c$ coloring of
$\binom{\N}{2}$, there is an infinite  homogenous set USING induction on $c$.
\item
(2 points)
Which proof do you like better?
Which one do you think gives better bound when you
finitize it?
\end{enumerate}


\vfill

\centerline{\bf GOTO NEXT PAGE}

\newpage

\item (20 points) Prove the following theorem rigorously (this is the
  infinite $c$-color $a$-ary Ramsey Theorem):

\noindent
{\bf Theorem}
    For all $a\ge 1$, for all $c\ge 1$, and for all $c$-colorings of
    $\binom{\mathbb{N}}{a}$, there exists an infinite set
    $A\subseteq \mathbb{N}$ such that $\binom{A}{a}$ is monochromatic
    ($A$ is an infinite homogeneous set).

\noindent
{\bf End of Statement of Theorem}

The proof should be by induction on $a$ with the base cases being $a=1$.
You need to prove the base case. 


\vfill

\centerline{\bf GOTO NEXT PAGE}

\newpage

\item
(20 points) Lets apply Ramsey Theory! 


\begin{enumerate}
\item 
(20 points) 
Let 

$$x_1,x_2,x_3,\ldots,$$

be an infinite sequence of distinct reals. 

Consider the following coloring of $\binom{N}{2}$. 
Let $i<j$. 

\begin{equation}
COL(i,j) =
\begin{cases}
RED & \hbox{if $x_i<x_j$}\cr
BLUE& \hbox{if $x_i>x_j$}\cr
\end{cases}
\end{equation}

If you apply Ramsey Theory to this coloring you get a theorem.

State that theorem cleanly. 

\item
(0 points, but REALLY try to do it)
Prove the theorem you stated in Part a WITHOUT USING Ramsey Theory.  

\item
(0 points, but REALLY do it) 
Which proof do you prefer, the one that use Ramsey Theory or the one that didn't? 
\end{enumerate}

\vfill

\centerline{\bf GOTO NEXT PAGE}

\newpage

\item
(20 points) Lets apply Ramsey Theory! 


\begin{enumerate}
\item 
(9 points) 
Let 

$$x_1,x_2,x_3,\ldots,$$

be an infinite sequence of points in $\R^2$. 
(NOTE- these are points in $\R^2$, not reals. So this is
a different setting from the prior problem.) 

Consider the following coloring of $\binom{N}{2}$. 

\begin{equation}
COL(i,j) =
\begin{cases}
RED & \hbox{if $d(x_i,x_j)>1$}\cr
BLUE& \hbox{if $d(x_i,x_j)<1$}\cr
\end{cases}
\end{equation}

If you apply Ramsey Theory to this coloring you get a theorem.
State that theorem cleanly. 

\item 
(9 points) 
Prove the theorem you stated in Part a WITHOUT USING Ramsey Theory.  

\item
(2 points) 
Which proof do you prefer? 
\end{enumerate}


\vfill

\centerline{\bf GOTO NEXT PAGE}

\newpage



\item
(Extra Credit- NOT towards your grade but towards a letter I may one day write for you)

{\it Definition} A {\it bipartite} graph is a graph with vertices $A\cup B$
and the only edges are between vertices of $A$ and vertices of $B$. 
$A$ an $B$ can be the same set. We denote a biparatite graph with a 3-tuple
$(A,B,E)$. 

{\it Notation} $K_{n,m}$ is the bipartite graph $([n],[m],[n]\times [m])$. 

{\it Notation} $K_{\N,\N}$ is the bipartite graph $(\N,\N,\N \times\N)$. 

{\it Definition} If $COL$ is a $c$-coloring of the edges of $K_{\N,\N}$
then $(H_1,H_2)$ is a homog set if $c$ restricted to $H_1\times H_2$ is
constant. 

And now FINALLY the problem. 

Prove or disprove:

{\it For every 2-coloring of the edges of $K_{\N,\N}$ there exists $H_1$, $H_2$ infinite
such that $(H_1,H_2)$ is a homog set. }

\newpage

\item
(Extra Credit- NOT towards your grade but towards a letter I may one day write for you)
Recall that the infinite Ramsey Theorem for 2-coloring the edges of a graph:

{\it For all colorings $COL:\binom{\N}{2}\into [2]$ there exists
an infinite homogenous set $H\subseteq \N$.}

What if we color $\Z$ instead of $\N$? If all we want is an {\it infinite homogenous set} then
the exact same proof works---or you could just restrict the coloring to $\binom{\N}{2}$. 
But what if we want an infinite $H\subseteq \Z$ that has {\it the same order type as $\Z$}?

{\bf Definition} If $(L_1,<_1)$ and $(L_2,<_2)$ are ordered sets then they are {\it order-equivalent} 
if there is a bijection $f$ from $L_1$ to $L_2$ that preserves order. That is, $x<_1 y$ iff $f(x) <_2 f(y)$. 

And now FINALLY the problem:

Prove or disprove: 

{\it For all colorings $COL:\binom{\Z}{2}\into [2]$ there exists
a set $H\subseteq \Z$ that is order-equiv to $\Z$ and is homogenous.}

\end{enumerate}
\end{document}