\documentclass[12pt,ifthen]{article}
\usepackage{amsmath}
\usepackage{amssymb}
%\usepackage{html}
\usepackage{hyperref}
\newcommand{\degg}{{\rm deg}}
\newcommand{\COL}{{\rm COL}}
\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 04} 

\centerline{Morally Due Tue Feb 22 at 3:30PM. Dead Cat Feb 24 at 3:30}

\begin{enumerate}

\item
(0 points)
What is your name?  Write it clearly.
When is the take-home midterm due?

\item
(40 points)
Assume $(X,\preceq_X)$ and $(Y,\preceq_Y)$ are wqo. 
Consider the ordering $(X\times Y,\preceq)$ where $\preceq$ is defined as

$$(x_1,y_1) \preceq (x_2,y_2) \hbox{ iff } x_1 \preceq_X x_2 \hbox{ AND } y_1\preceq_Y y_2.$$

Show that $(X\times Y,\preceq)$ is a wqo.

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\newpage

\item
(50 points)
Assume $(X,\preceq)$ is a wqo. 
Let $\PF(X)$ be the set of finite subsets of $X$. 
Let $\preceq'$ be the following order on $\PF(X)$. 

Let $Y,Z\in \PF(X)$. 

$Y\preceq' Z$ iff

there exists a FUNCTION $f:Y\into Z$ such that $(\forall y\in Y)[y\preceq f(y)]$.

\begin{enumerate}
\item
(20 points)
Prove or disprove: $(\PF(X),\preceq')$ is a wqo. 
\item 
(15 points)
Modify $\preceq'$ such that the function $f$ has to be injective (also called 1-1). 
Prove or disprove: $(\PF(X),\preceq')$ is a wqo. 
\item
(15 points)
Modify $\preceq'$ such that the function $f$ has to be surjective (also called onto). 
Prove or disprove: $(\PF(X),\preceq')$ is a wqo. 
\end{enumerate}

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\newpage

\item 
(10 points) GOTO my webpage of funny music and GOTO  the section on Math Songs

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

\begin{itemize}
\item 
Listen to the Bolzano Weirstrauss rap- or as much of it as you can stand. 
Comment on it.
\item 
Pick ANY OTHER math song AT RANDOM and listen to it. Is it better than the BW rap
(hint: YES). Comment on it. 
\end{itemize} 

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\newpage



\item
(Extra Credit- NOT towards your grade but towards a letter I may one day write for you)
(This will look like a prior extra credit but it's a new problem.) 

{\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 $B$. 
$A$ and $B$ can be the same set. We denote a bipartite 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 $H_1\times H_2$ takes
on only 1 value (I changed the wording on this so I can generalize it later.) 

{\bf RECALL} In a prior extra credit problem we DISPROVED the following:

{\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. }

In other words we showed the following:

{\it There IS a 2-coloring of the edges of $K_{\N,\N}$ such that there is NO $H_1$, $H_2$ infinite
such that $(H_1,H_2)$ is a homog set. }

This inspires the following definition.

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

SO to recap- we could have a 2-coloring of the edges of $K_{\N,\N}$ where there is no 1-homog set. 
But there is clearly a 2-homog set, namely $(\N,\N)$. 

{\bf And now FINALLY the problem:}

{\it For ever $k\ge 3$ 
Prove or disprove:
For every $k$-coloring of the edges of $K_{\N,\N}$ there exists $H_1$, $H_2$ infinite
such that $(H_1,H_2)$ is a 2-homog set. 
}

\end{enumerate}
\end{document}

