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\centerline{\bf Homework 2} 
\centerline{\bf Morally Due Tue Feb 11, 2025, at 3:30PM} 
\centerline{\bf Dead Cat Feb 13, 2025, at 3:30PM} 
\bigskip

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\begin{enumerate}

\item
(30 points) 

Recall: If $L_1,L_2$ are two ordered sets then $L_1\equiv L_2$ means there is an
order-preserving bijection between $L_1$ and $L_2$. 

Let $5\omega$ be $\omega+\omega+\omega+\omega+\omega$. 

This ordering is five copies of $\omega = \{ 1<2<3<\cdots\}$ concatenated. 

Fill in the value of $c$ in the next two statements (the same value) and
prove both parts. You may use the infinite Ramsey Theorem and the Infinite Bipartite Ramsey Theorm.

\begin{itemize}
\item
There exists $\COL \colon \binom{5\omega}{2}\into [c]$ such that there is NO  $(c-1)$-homog $H\equiv 5\omega$. 
\item
For all $\COL \colon \binom{5\omega}{2}\into [1,000,000]$ there exists a $c$-homog $H\equiv 5\omega$. 
\end{itemize} 

\newpage

\item
(30 points)

Recall: If $L_1,L_2$ are two ordered sets then $L_1\equiv L_2$ means there is an
order-preserving bijection between $L_1$ and $L_2$. 

Let $\omega^2$ be $\omega+\omega+\cdots$. 

Fill in the value of $c$ in the next two statements (the same value) and
prove both parts. You may use the infinite Ramsey Theorem and the Infinite Bipartite Ramsey Theorm.

\begin{itemize}
\item
There exists $\COL \colon \binom{\omega^2}{2}\into [c]$ such that there is NO  $(c-1)$-homog $H\equiv \omega^2$. 
\item
For all $\COL \colon \binom{\omega^2}{2}\into [1,000,000]$ there exists a $c$-homog $H\equiv \omega^2$. 
\end{itemize} 

\newpage

\item
(40 points)

In class we proved the following:

{\it The Infinite Ramsey Theorem},

{\it The Finite Ramsey Theorem with a proof that gave no bounds on $R(k)$} 

{\it The Finite Ramsey Theorem with a proof that gave the bound $R(k)\le 2^{k-1}$}
This last proof followed the proof of the {\it Infinite Ramsey Theorem} closely. 

\bigskip

We then proved

{\it Infinite 3-hypergraph Ramsey Theorem}

YOUR ASSIGNMENT:

{\it Proof The Finite Ramsey Theorem with a proof that gave a bound on $R(k)$. }

This prove should follow the proof of the {\it Infinite 3-hypergraph 
Ramsey Theorem} closely. 

Make sure to state what the upper bound you get is. 

{\bf Hint} The proof of the 2-ary Ramsey Theorem used the 1-ary Ramsey Theorem.
You will need the following restatement of 2-ary 

{\it For all $\COL\colon\binom{[n]}{2}\into[2]$ there exists a homog set of size $\ge 0.5\log n$. }


\end{enumerate} 


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