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\begin{document}

\centerline{\bf Homework 02, Morally Due 12:30PM, Tue Feb 10,  2026}

\newif{\ifshowsoln}
%\showsolntrue % comment out to NOT show solution inside \ifshowsoln and \fi blocks.

\begin{enumerate}

\item
(0 points) What is your name. What is your quest? What is your favorite color?
What are those three questions from (do not look it up, I want to know how many know honestly). 

\bigskip

This HW has 4 questions and 1 extra credit question

\vfill
\eject 


\item
(25 point) We take $\Z$ to be $\{\cdots, -3< -2< -1< 1 < 2 <3 \cdots \}$. 

Let $\COL \binom{Z}{2}\into [3]$ be defined as follows:

Assume $x<y$. 

\begin{equation}
\COL(x,y)=
\begin{cases}
\R & \hbox{if $x,y\ge 1$ } \\
\B & \hbox{if $x,y\le -1$ } \\
\G & \hbox{if $x\le -1$, $y\ge 1$} \\
\end{cases}
\end{equation}

Show that there is no 2-homog set $H \equiv Z$. 

\vfill\eject

\item 
(25 point) We take $\Z$ to be $\{\cdots < -6< -4< -2< 1 < 3 <5< \cdots \}$. 

Let $\COL \binom{Z}{2}\into [4]$ be defined as follows:

We assume $|x|<|y|$. 

\begin{equation}
\COL(x,y)=
\begin{cases}
1 & \hbox{if $x,y\ge 1$ } \\
2 & \hbox{if $x,y\le -1$ } \\
3 & \hbox{if $x\le -1$, $y\ge 1$} \\
4 & \hbox{if $y\le -1$, $x\ge 1$} \\
\end{cases}
\end{equation}

Show that there is no 3-homog set $H\equiv \Z$. 

\vfill\eject

\item 
(25 points)
Find $X\in\N$ such that the following two statements hold.

\begin{itemize}
\item
 $\exists$ $\COL\colon \binom{\omega+\omega+\omega}{2}\into [X]$ such that there is no $(X-1)$-homog $H\equiv \omega+\omega+\omega$. 

\bigskip

\item 
$\forall d$ $\forall $ $\COL\colon \binom{\omega+\omega+\omega}{2}\into [d]$ $\exists$ $X$-homog $H\equiv \omega+\omega+\omega$. 
\end{itemize}

\vfill\eject

\item 
(25 points)
Let $n\in\N$.
The notation $n\omega$ is the ordering

$$\omega + \omega + \cdots + \omega$$ 

There are $n$ copies of $\omega$.

\bigskip

Find $X(n)\in\N$ such that the following two statements hold.

\begin{itemize}
\item
 $\exists$ $\COL\colon \binom{n\omega}{2}\into [X(n)]$ such that there is no $(X(n)-1)$-homog $H\equiv n\omega$. 

\bigskip

\item 
$\forall d$
$\forall$ $\COL\colon \binom{n\omega}{2}\into [d]$ $\exists$ $X(n)$-homog 
$H\equiv n\omega$. 
\end{itemize}

\vfill\eject

\item
(Extra Credit)
(This problem is not related to the material covered so far.) 
You have to do both parts.

Let $n\in\N$. The number $n$ is {\it jiggy}
if there exists a finite collection of sets 
$A_1,\ldots,A_m$ such that the following hold:

\begin{itemize}
\item 
For all $1\le i \le m$, $A_i \subseteq \{1,\ldots,n\}$. 
\item
For all $1\le i \le m$, $|A_i|=5$.
\item
For all $1\le i<j\le m$, $|A_i\cap A_j|=1$.
\item
For all $1\le k\le n$, there exists $i$, $k\in A_i$.
\end{itemize}

This Extra Credit is in two parts. You'll see why.


PART I: Morally due with the HW, Feb 10 at 12:30PM.
\begin{enumerate}
\item
Find an infinite set of $n$ that are jiggy. 
\item
Find an infinite set of $n$ that are not jiggy.
\end{enumerate}

\bigskip

Part II: Due by the end of the semester:
Characterize exactly which numbers are jiggy. 
\end{enumerate}

\end{document}

