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\centerline{\bf HW 01 CMSC 752}
\centerline{\bf Morally Due Feb 6 3:30PM. Dead-Cat Due Feb 8 at 3:30PM} 

\noindent
{\bf Course Website}

\url{https://www.cs.umd.edu/~gasarch/COURSES/752/S25/index.html}

\begin{enumerate}
\item
(0 points but do it anyway.)
\begin{enumerate}
\item
If you are not on piazza, put yourself on piazza.
\item
If you are not on gradescope, put yourself on gradescope.
The Entry Code is DKW6VX. 
(The school automatically put everyone enrolled in the class on
gradescope but it might have messed up, and some of you might not be
on the roster yet.)
\item 
(0 points)
Learn LaTeX and write some simple documents in it. 
\item
(0 points) 
Learn the package tikz that you can use to draw pictures in LaTeX. 
\end{enumerate}

\newpage

\item
(50 points) 
{\it Let $p_1,p_2,\ldots$ be an infinite set of distinct reals.
Show that at least one of the following is true:
\begin{itemize}
\item
There exists an infinite increasing subsequence

(so there exists $i_1< i_2 < \cdots $such that 

$p_{i_1} < p_{i_2} < \cdots$.) 

\item 
There exists an infinite decreasing subsequence

(so $i_1< i_2 < \cdots $such that 

$p_{i_1} > p_{i_2} > \cdots$.) 

\end{itemize}
}
{\bf Hint:} Consider the coloring 

$\COL\colon \binom{\N}{2}\into\{INC,DEC\}$ defined by

(assume $i<j$) 

\begin{equation} 
\COL(i,j)= 
\begin{cases}
INC & \text{ if $p_i<p_j$}\\
DEC & \text{ if $p_i>p_j$}\\
\end{cases}
\end{equation} 

\newpage 

\item
\begin{enumerate}
\item 
(0 points but do it) 
The theorem in Part 1 is a Lemma in the proof of the Bolzano-Weierstrass theorem.
See the proof on Wikipedia

\url{https://en.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass_theorem}


Look at the proof on Wikipedia.
Is it the same proof as you gave in Part 1, only messier?
Or is it a different proof? 

\item
(0 points but DO IT) 

Listen to The Bolzano Weierstrass Rap:

\url{https://www.youtube.com/watch?v=dfO18klwKHg}

Then list to any three other RANDOM math songs from my collection,
which is here: 

\url{https://www.cs.umd.edu/~gasarch/FUN/mathsongs.html}

Which was your favorite song?

Which was your least favorite song?

Was your least favorite song better than the BW rap
\end{enumerate} 

\newpage 

\item
  (50 points)
(This is not a question in Ramsey Theory but it is a prelude to one of our
``Applications.'')

{\bf Notation} $\Z$ is the integers. $\Z[x_1,\ldots,x_n]$ is the
set of all polynomials in $\{x_1,\ldots,x_n\}$ (they do not have to all
appear) with coefficients in $\Z$. 

Let $p(x,y)\in\Z[x,y]$.

We view $p$ as a function from $Z\times\Z$ into $\Z$. 

Note that $p(x,y)$ might be onto (e.g., $p(x,y)=x+y+1$) 
or not (e.g., $p(x,y)=x^2 + y^2+1$). 

Show there is an infinite set $\D\subseteq \Z$ such that $p$ restricted to $\D\times \D$
is NOT onto. 

\newpage

\item
(EXTRA CREDIT) 
Let $q(x,y)=\ceil{p(x,y)^{1/100}}$. 

Show there is an infinite set $\D\subseteq \Z$ such that $q$ restricted to $\D\times\D$
is NOT onto. 

\item
(EXTRA CREDIT)
Prove or Disprove: For ALL function $p:\colon \Z\times\Z \into \Z$ there is 
an infinite set $\D\subseteq\Z$ such that $p$ restricted to $\D\times \D$ is NOT onto. 


\end{enumerate}


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\end{enumerate}
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