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\begin{document}

\centerline{\bf Homework 12, Morally Due 12:30PM, Tue May 5 2026}


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

\begin{enumerate}

\item
(0 points) What is your name? 


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\newpage

\item
(30 points) 

\begin{enumerate}
\item
(0 points)
Write a program that, on input $(n,k,c)$, looks at ALL $c$-colorings
of $[n]$, counts how many HAVE a $k$-AP, and divides that by $c^n$.
Hence you get the fraction of $c$-colorings that have a mono $k$-AP to 3 places. 

\item
(15 points) 
For $3\le n\le 7$ determine what fraction of 2-colorings of $[n]$ have a mono 3-AP. 
Output a table like the one below. THE NUMBERS WE GIVE ARE FALSE or true by accident.

\begin{tabular}{|c|l|} 
\hline
$n$ & Fraction of 2-colorings of $[n]$ that have a mono 3-AP \cr 
\hline
3   & 0.5 \cr
4   & 0.617 \cr
5   & 0.689 \cr
6   & 0.75 \cr
7   & 0.81 \cr
8   & 0.999 \cr
\hline
\end{tabular} 

\bigskip

\item
(15 points)
For $3\le n\le 20$ determine what fraction of 2-colorings of $[n]$ have a mono 4-AP. 
Output a table similar to the one in the last problem.

\bigskip

\item
(0 points)
Are the numbers surprising (e.g., more or less than you would have thought)? 

\end{enumerate}

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\item 
(40 points. 
This problem is also on the optional project so if you are doing the optional
project,  then use your answer for both this HW and the project. DO NOT say on the optional
project {\it see my answer on HW12}). 

Let $VDW(k,\omega)$ mean that VDW theorem is known for $k$-APs and ANY number of colors. 

\bigskip

Assume you know VDW theorem for $VDW(1,\omega)$, $VDW(2,\omega)$, $VDW(3,\omega)$, 
and $VDW(4,\omega)$.
Prove VDW theorem for $VDW(5,2)$. 

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\item
(30 points. 
\bigskip

You cannot Use Rado's Theorem for this problem.
We will be asking you to prove Rado's Theorem in this particular
case. 
You CAN use the Extended VDW theorem. 

Consider the equation 

$$E: \\ v+ 2w - 3x + y + 10z =0.$$

Show that there exist a number $M$ such that, for all $\COL\colon [M]\into [2]$,
there exists a mono solution to $E$ (The $v,w,x,y,z$ might not be distinct.) 

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\item
(0 points. AND YOU MUST DO THIS BY MONDAY SO THAT I CAN TALK ABOUT IT TUESDAY.) 

What is funniest thing you saw in class. Here are options though you are not limited to them.

\begin{enumerate}
\item
The Bolzano Weisterauss Rap

\item
The repeated sketch where Ryan's Friend and I changed shirts.

\item
The Banach Tarski Paradox used to explain the miracle of loaves and fishes.

\item
The Combinatorics Song

\item
Bill's Rap about the Infinite Hat Game. 

\item
BAD SEQUENCES! Song about the infinite hat game. 

\item
Bill sold the rights to the song to Who? 

\item
Celebrating Ryan's friend getting his Masters

\item
{\it That is what Erdos calls it The Happy Ending Theorem.

Their marriage lasted 68 years and they died on the same day.

We prove this 3 ways.}

\item
The Application of Ramsey Theory to History. 

\item
Quantum Supremacy vs you-know-what. 

\end{enumerate}

I am sure there are others. For example, 

\begin{enumerate}
\item 
In the past one of the students
said that the poly VDW theorem was funny since the bounds
were so big.  I didn't do TREE(3) that year which she would have
thought was hilarious. 

\item 
Some students have pointed to a line or part of a sketch. 

\end{enumerate}











\end{enumerate}

\end{document}

