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\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 infinite 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
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\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 $ix_j$}\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
GREEN& \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}