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\centerline{\bf CMSC 752 Homework 4} 
\centerline{\bf Morally Due Tue Feb 25, 2025} 
\centerline{\bf Dead Cat Feb 27}

\bigskip

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

{\bf Notation} $R_a(k)$ is the $a$-ary Ramsey number. 

\begin{enumerate}
\item
(30 points) 
RECALL:

We used $R_2(k)\le 2^{2k}$ to obtain $R_3(k)\le 2^{2^{4k}}$. 

\bigskip

What if Arushi, Danesh, and Sharma show $R_2(k)\le 2^{1.9k}$?

What if Soren and Rishi show $R_2(k)\le 2^{1.7k}$?

What if Larry, Moe, and Curly show $R_2(k)\le 2^{1.5n}$? 

\bigskip

How will these new bounds on $R_2(k)$ help get better bounds on
$R_3(k)$?

That is the point of this problem. 

\bigskip

Find a function $f$ that has the following properties (and prove them):

\begin{itemize}
\item
$f(2)=4$
\item
If $x<2$ then $f(x)<4$. 
\item 
{\it IF $R_2(k)\le 2^{ck}$ THEN $R_3(k) \le 2^{2^{f(c)k}}$}.

(Note: You may get an upper bound like (I am making this up) $2^{2^{3n+7}}$.
Ignore the additive contant 7.) 

\bigskip 
I am asking you to reprove the finite 3-hypergraph Ramsey Theorem
with the parameter $c$ so that you can see how an improvement on the upper bound for $R_2(k)$
will result in an improvement on the upper bound for $R_3(k)$.
\end{itemize} 

\newpage 

\item
(30 points) 
Find a function $f$ that has the following properties (and prove them):

\bigskip

{\it IF $R_3(k)\le 2^{2^{ck}}$ THEN $R_4(k) \le f(c,k)$} 

\bigskip

I am asking you to prove the finite 4-hypergraph Ramsey Theorem
with the parameter $c$ so that you can 

(a) see what the upper bound on $R_4(k)$ was when the upper bound on $R_3(k)$ was $2^{2^{4k}}$. 

(b) see what the upper bound on $R_4(k)$ was now that the upper bound on $R_3(k)$ was $2^{2^{2k}}$. 

(c) be prepared for the future when Danesh-1, Danesh-2, and Danesh-3 team up to show 
$R_3(k)\le 2^{2^{1.99k}}$. 

\newpage

\item
(40 points) 
\begin{enumerate}
\item 
(40 points) 
Prove the following: For all $n\ge 2$, for all functions 

$$f\colon \Z^n\into \Z$$

there exists infinite $\D\subseteq\Z$ such that

$$f\colon \D^n\into \Z$$

is NOT onto. 

\item
(0 points) THINK ABOUT:  Let $T(n)$ be the number of times you used
Ramsey's Theorem in the proof of Part 1.  What was your $T(n)$?

\end{enumerate} 


\newpage
\item
(0 points, Extra Credit)
\begin{enumerate}
\item
Give your name (this will not get you any extra credit, but since I
grade this one by putting the names of who got it right into a file,
this makes my life easier.)

\item 
Recall that a coloring of the edges of $K_{n,n}$ (the complete bipartite graph)
is a coloring of $[n]\times [n]$. 


Prove the following 

{\it For all $k$ there exists $n=B(k)$ such that for all 

$$\COL\colon [n]\times[n] \into [2]$$

there exists $A\subseteq [n]$ and $B\subseteq[n]$ such that
$|A|=|B|=k$ and $\COL$ restricted to $[k]\times[k]$ is constant. 
Provide bounds on $B(k)$. 
}
\end{enumerate} 

\end{enumerate} 


\end{document}