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\begin{document}

\centerline{\bf Ramsey Theory Project. Morally Due 12:30PM May 5, 2026}



\begin{enumerate}

\item
(0 points) What is your name? 


\vfill

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\newpage

\item 
(10 points) 
For this problem you can assume that $R(k)=2^k$. 

\begin{enumerate}
\item 
Fill in the following sentence and prove it:

\bigskip

{\it Let $k\in\N$. If $n$ satisfies condition $XXX(k)$ then, for all 
$\COL \colon \binom{[n]}{2}\into [2]$, 
there exists at least 3  mono $K_k$'s. 

\bigskip

Your proof  should use a similar technique used in class for $K_4$. }

\item 
(0 point and don't hand in but this will help you with Part 3)
Write a program that will, given $L$, find the least $n$ that satisfies $XXX(L)$. 

\item
(5 points) 
Run the program for $k=4$ to 7 give us a table of the following form
(I only give the first three rows and the numbers are made up.) 

\[
\begin{array}{|r|l|}
\hline
k & \hbox{least n} \cr 
\hline
4 & 22 \cr
5 & 27 \cr
6 & 37 \cr
7 & 47 \cr
\hline
\end{array}
\] 

\item
(5 points)
Based on your data, come up with a conjecture for what $n$ is as a function of $k$. 

\end{enumerate} 

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\item
(20 points) 
For this problem you may use the following theorem we proved in class:

\noindent
$\forall$ $\COL\colon \N\times\N \into [10^{100}]$
$\exists$  infinite $A,B\subseteq \N$ such that 

$\COL \colon  A\times B \into [10^{100}]$ only uses $2$ colors. 

\bigskip

Fill in the $c$ and prove the resulting theorem:
(By `prove' I mean show me the first few steps of the construction
and hand-wave the rest.) 


\bigskip

{\it 

\noindent
1) $\forall$ $\COL\colon \N\times\N\times \N \into [10^{100}]$
$\exists$  infinite $A,B,C\subseteq \N$ such that 

$\COL \colon  A\times B\times C  \into [10^{100}]$ only uses $c$ colors. 


\bigskip

\noindent
2) $\exists$ $\COL\colon \N\times\N\times\N \into [c]$ such that
there is no infinite $A,B,C\subseteq\N$ such that 

$\COL \colon  A\times B\times C \into [c]$ only uses $c-1$ colors.

}


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\item 
(20 points) 
Prove that the following function exists:

\bigskip

$f(n)$ is the least $m$ such that there for every sequence of graphs

$$G_1,G_2,\ldots,G_m$$ 

where the number of vertices in $G_i$ is $\le i+n$

there is an $i<j$ with $G_i\preceq_{\rm minor} G_j$. 

You may use the Graph Minor theorem. 


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\item 
(20 points. As of April 14 I have not gone over the material for this problem)
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 

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

Show that there exist a number $M$ such that, for all $\COL\colon [M]\into [2]$,
there exists a mono solution to $E$ with $w,x,y,z$ all distinct. 


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\item 
(20 points. As of April 14 I have not gone over the material for this problem)

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 $VDW(5,2)$. 



\end{enumerate}


\end{document}

