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\begin{document}

\centerline{\bf Homework 08, Morally Due 12:30PM, Tue Apr 7  2026}


\newif{\ifshowsoln}
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\begin{enumerate}

\item
(0 points) What is your name? 


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\item
(30 points)
Let $\COL \colon \binom{[n]}{2} \into [\omega]$.
Assume that, for all colors $c$, for all $x\in [n]$, 
$\deg_c(x)\le 17$. Show that there is a rainbow set of size 
$\Omega(n^{1/3})$. 


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\item
(40 points) 
{\bf Recall:} we showed  the following:

For all $f\colon \Z\times \Z \into \Z$
there exists infinite $\D\subseteq \Z$ such that 

$f$ restricted to $\D\times\D$ is NOT onto. 

Recall that we used the 1-ary Ramsey Theorem  ONCE
and the 2-ary Ramsey Theorm TWICE.
We could that as THREE uses of Ramsey's Theorem. 

And NOW for our problems

\begin{enumerate}
\item 
Let $f\colon \Z\times \Z \times \Z \into \Z$. 

Show there exists a $\D\subseteq \Z$ such that 

$f$ restricted to $\D\times\D\times\D$ is not onto. 

How many times does your proof use the Ramsey's theorem?

\item
Let $a\in\N$. 

Let $f\colon \Z^a\into \Z$. 

Show there exists a $\D\subseteq \Z$ such that 

$f$ restricted to $\D^a$ is not onto. 

Let $g(a)$ be the number of times your proof uses the Ramsey's theorem.

If you did the problem correctly then $g(a)$ does not have a nice closed form
but it is the solution to a combinatorial problem. What is that problem? 
\end{enumerate}


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\item
(30 points) 
{\bf Recall} that a finite set $X$ is {\it Large} if $|X|>\min(X)$. 

Prove the following: 

{\it For all $k$, there exists an $n$, such that, for all 
$\COL\colon \binom{\{k,k+1,\ldots,k+n\}}{2}\into [\omega]$ 
there is a large set that is either 
\begin{itemize}
\item
homog
\item
min-homog
\item
max-homog
\item
rainbow
\end{itemize}
}

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\item
(0 points, but you must answer it)

As you all know, Yasmine's stage name is {\it Bad Sequence}.

If I had a rock band I would call it {\it The Red Cliques}.

A grad student in quantum computing had a group called {\it Amplitudes with Attitude}. 

Dave Mount (Geometry) would name his group {\it Fried Green Ellipsoids}. 

The following are real groups:

{\it The Mobius Band},

{\it The Klein Four},

{\it The MIT Logarhythms}.

There is a rap singer named MC++. 
There is subgenre of rap called {\it Nerdcore} where people sing about math and technology things.
{\it Drama in the PhD} is the best example of that. Do not confuse this with {\it mathcore} which is
{\it a highly technical, chaotic subgenre of hardcore punk and metalcore} which in my opinion is
just noise.

\bigskip

OKAY, so finally here is the question: 

If you had a stage name, what would it be?

If you had a rock group, what would you name it? 


\end{enumerate}

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